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# The rule of conditional probability
is valid in quantum theory

P.G.L. Porta Mana  Kavli Institute, Trondheim <portamana.org> pgl

(13 July 2020; updated 31 July 2020)

In a recent manuscript, Gelman & Yao *gelmanetal2020 claim that “the usual
rules of conditional probability fail in the quantum realm” and that
“probability theory isn’t true (quantum physics)”, and purport to support
these statements with the example of a quantum double-slit experiment. Their
statements are false. In fact, opposite statements can be made, from two
different perspectives:

  * •

The example given in that manuscript confirms, rather than invalidates, the
probability rules. The probability calculus shows that a particular relation
between probabilities, to be discussed below, _cannot a priori_ be assumed to
be an equality or an inequality. In the quantum example it turns out to be an
inequality, thus confirming what the probability calculus says.

  * •

But actually the same inequality can be shown to appear in very non-quantum
examples, such as drawing from an urn. Thus there is nothing peculiar to
quantum theory in this matter.

In the present comment I will prove the two points above, recalling some
relevant literature in quantum theory. I shall also correct a couple of wrong
or imprecise statements that Gelman & Yao make about quantum physics in their
example.

Let me point out at the outset that the rules of probability theory (product
or conditional or conjunction, sum or disjunction, negation) are in fact
routinely used in quantum theory with full validity, especially in problems of
state “retrodiction” and measurement reconstruction [jones1991b,
slater1995b][chs
7,8]demuynck2002b_2004barnettetal2003,zimanetal2004_r2006,darianoetal2004[see][§
1 and the rest of the present comment for many further
references]maanssonetal2006. An example is the inference of the state of a
quantum laser given its output through different optical apparatus
[leonhardt1997].

Similar incorrect claims with similar examples have appeared before in the
quantum literature [brukneretal2001, see e.g.]. Bernard O. Koopman [[, of the
Pitman-Koopman theorem for sufficient statistics,]]koopman1936 discussed the
falsity of such claims already in [*]koopman1957. The Introduction in his work
is very clear:

> Ever since the advent of modern quantum mechanics in the late 1920’s, the
> idea has been prevalent that the classical laws of probability cease, in
> some sense, to be valid in the new theory. More or less explicit statements
> to this effect have been made in large number and by many of the most
> eminent workers in the new physics […]. Some authors have even gone farther
> and stated that the formal structure of logic must be altered to conform to
> the terms of reference of quantum physics […].
>
> Such a thesis is surprising, to say the least, to anyone holding more or
> less conventional views regarding the positions of logic, probability, and
> experimental science: many of us have been apt – perhaps too naively – to
> assume that experiments can lead to conclusions only when worked up by means
> of logic and probability, whose laws seem to be on a different level from
> those of physical science.
>
> The primary object of this presentation is to show that the thesis in
> question is entirely without validity and is the product of a confused view
> of the laws of probability.

It must be remarked that such claims have hitherto never been supported by any
rigorous proof – with explicitly stated definitions and assumptions, well-
defined and unambiguous notation, and clear logical and mathematical steps
(and Gelman & Yao are no exception). The typical fallacy in the kind of
examples presented rests in the neglect of the experimental setup, leading
either to an incorrect calculation of conditional probabilities, or to the
incorrect claim that the probability calculus yields an equality, where it
actually does not. The same incorrect claims can be obtained _with completely
non-quantum systems_ , such as drawing from an urn, if the setup is neglected
[kirkpatrick2001_r2003, kirkpatrick2002_r2003][§ IV]portamana2003_r2004.

Let us start with such a non-quantum counter-example.

## A non-quantum counter-example

Consider an urn with one $B$lue and one $R$ed ball. Two possible drawing
setups are given:

  1. ${\color[rgb]{0.265625,0.46484375,0.66796875}D_{\textrm{a}}}$

With replacement for blue, without replacement for red. That is, if blue is
drawn, it is put back before the next draw (and the urn is shaken); if red is
drawn, it is thrown away before the next draw.

  2. ${\color[rgb]{0.66796875,0.19921875,0.46484375}D_{\textrm{b}}}$

With replacement for red, without replacement for blue.

These two setups are obviously mutually exclusive.

We can easily find the unconditional probability for blue at the second draw
in the setup ${\color[rgb]{0.265625,0.46484375,0.66796875}D_{\textrm{a}}}$:

$\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.265625,0.46484375,0.66796875}D_{\textrm{a}}})=\tfrac{3}{4}\;.$
(1)

Note that this probability can be intuitively found by simple enumeration, à
la Boole, considering “possible worlds” if you like. Out of four possible
worlds, half of which has blue at the first draw, and the other half has red,
we can count that three worlds have blue at the second draw.

The conditional probabilities for blue at the second draw, given the first
draw, are also easily found:

$\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}B_{1}\land{\color[rgb]{0.265625,0.46484375,0.66796875}D_{\textrm{a}}})=\tfrac{1}{2}\qquad\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}R_{1}\land{\color[rgb]{0.265625,0.46484375,0.66796875}D_{\textrm{a}}})=1\;.$
(2)

We find that

$\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.265625,0.46484375,0.66796875}D_{\textrm{a}}})=\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}B_{1}\land{\color[rgb]{0.265625,0.46484375,0.66796875}D_{\textrm{a}}})\;\mathrm{P}(B_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.265625,0.46484375,0.66796875}D_{\textrm{a}}})+\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}R_{1}\land{\color[rgb]{0.265625,0.46484375,0.66796875}D_{\textrm{a}}})\;\mathrm{P}(R_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.265625,0.46484375,0.66796875}D_{\textrm{a}}})\;,$
(3)

which is just the rule of conditional probability. It is in fact just the
systematization and generalization of the intuitive “possible worlds”
reasoning done above.

Next consider the setup
${\color[rgb]{0.66796875,0.19921875,0.46484375}D_{\textrm{b}}}$. We easily
find

$\displaystyle\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{\textrm{b}}})=\tfrac{1}{4}\;,$
(4)
$\displaystyle\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}B_{1}\land{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{\textrm{b}}})=0\qquad\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}R_{1}\land{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{\textrm{b}}})=\tfrac{1}{2}\;,$
(5)
$\displaystyle\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{\textrm{b}}})=\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}B_{1}\land{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{\textrm{b}}})\;\mathrm{P}(B_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{\textrm{b}}})+\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}R_{1}\land{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{\textrm{b}}})\;\mathrm{P}(R_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{\textrm{b}}})\;.$
(6)

Now compare the unconditional probability for blue at the second draw in the
setup ${\color[rgb]{0.265625,0.46484375,0.66796875}D_{\textrm{a}}}$, with the
conditional probabilities for blue at the second draw given the first draw in
the setup ${\color[rgb]{0.66796875,0.19921875,0.46484375}D_{\textrm{b}}}$:

$\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.265625,0.46484375,0.66796875}D_{\textrm{a}}})\mathrel{\bm{\neq}}\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}B_{1}\land{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{\textrm{b}}})\;\mathrm{P}(B_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{\textrm{b}}})+\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}R_{1}\land{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{\textrm{b}}})\;\mathrm{P}(R_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{\textrm{b}}})\;.$
(7)

This inequality is not surprising – we are comparing different setups. It is
_not_ an instance of the conditional-probability rule. In fact the probability
calculus has nothing to say, a priori, about the relation between the left
side and right side, which are conditional on different statements or, if you
like, pertain to two different sample spaces.

You can call the inequality above “interference” if you want; for further and
more involved examples with urns and decks of cards see Kirkpatrick
*kirkpatrick2001_r2003,kirkpatrick2002_r2003 and Porta Mana *[§
IV]portamana2003_r2004.

Now consider another pair of drawing setups: setup
${\color[rgb]{0.1328125,0.53515625,0.19921875}D_{\textrm{c}}}$, with
replacement for both colours; and setup
${\color[rgb]{0.80078125,0.734375,0.265625}D_{\textrm{d}}}$, without
replacement for either colour. You can easily find that

$\displaystyle\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.1328125,0.53515625,0.19921875}D_{\textrm{c}}})=\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}B_{1}\land{\color[rgb]{0.1328125,0.53515625,0.19921875}D_{\textrm{c}}})\;\mathrm{P}(B_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.1328125,0.53515625,0.19921875}D_{\textrm{c}}})+\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}R_{1}\land{\color[rgb]{0.1328125,0.53515625,0.19921875}D_{\textrm{c}}})\;\mathrm{P}(R_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.1328125,0.53515625,0.19921875}D_{\textrm{c}}})\;,$
(8)
$\displaystyle\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.80078125,0.734375,0.265625}D_{\textrm{d}}})=\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}B_{1}\land{\color[rgb]{0.80078125,0.734375,0.265625}D_{\textrm{d}}})\;\mathrm{P}(B_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.80078125,0.734375,0.265625}D_{\textrm{d}}})+\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}R_{1}\land{\color[rgb]{0.80078125,0.734375,0.265625}D_{\textrm{d}}})\;\mathrm{P}(R_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.80078125,0.734375,0.265625}D_{\textrm{d}}})\;,$
(9)
$\displaystyle\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.1328125,0.53515625,0.19921875}D_{\textrm{c}}})\mathrel{\bm{=}}\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}B_{1}\land{\color[rgb]{0.80078125,0.734375,0.265625}D_{\textrm{d}}})\;\mathrm{P}(B_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.80078125,0.734375,0.265625}D_{\textrm{d}}})+\mathrm{P}(B_{2}\nonscript\,|\nonscript\;\mathopen{}R_{1}\land{\color[rgb]{0.80078125,0.734375,0.265625}D_{\textrm{d}}})\;\mathrm{P}(R_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.80078125,0.734375,0.265625}D_{\textrm{d}}})\;.$
(10)

The first two equalities above are expressions of the conditional-probability
rule. The third is _not_ , however. It is simply a peculiar equality
contingent on the two specific setups.

The probability calculus therefore correctly handles situations leading to
inequalities such as (7), and to equalities such as (10).

The explicit presence of ‘ $D_{\dotso}$ ’, which represents given information,
is necessary discussions involving different setups, such as the above. If I
ask you “what’s the probability of blue at the second draw?”, you will ask me
“in which drawing setup?”. The probability is conditional on the information
about the drawing scheme.

The inequality (7) is what Gelman & Yao *[p. 2]gelmanetal2020 complain about,
but in the context of a pair of quantum setups. I do not see how one can
complain about it, or claim inconsistencies. It is obviously correct even from
an intuitive analysis of the two setups. And the probability calculus
correctly leads to it, too. The probability calculus correctly leads also to
the equality (10). As already said, given two mutually exclusive setups, the
probability calculus a priori neither commits to an equality nor to an
inequality.

I will now show that the simple example above is in fact conceptually quite
close to the quantum experiment mentioned by Gelman & Yao. The closeness is
especially clear from the experimental and mathematical developments of
quantum theory of the past 40 years (at the very least), as the literature
cited below shows.

## The quantum double-slit experiments

The basic argument of Gelman & Yao is that, in a given setup of the quantum
double-slit experiment, we have a specific probability distribution for the
appearance of an emulsion or excitation on some point of the screen. We can
call this a “screen detection”, but please keep in mind that in so doing we
are adding an extra interpretation that modern quantum theory does not
actually commit to (see discussion and references below). In a different
experimental setup we have conditional probabilities for screen detection
conditional on slit detection. Now, the probability of the first setup is not
equal to the combination of the conditional probabilities of the second setup.

But this is exactly what happened in our urn example above, eq. (7). In the
present quantum case we do _not_ have a violation of the conditional
probability rule either – if anything it is a confirmation.

To see the analogy more clearly, let me present some additional facts from
quantum theory.

The experimental setup without detectors at the slits and the setup with slit
detectors are actually limit cases of a continuum of experimental setups
[woottersetal1979][for a recent review and further references
see][]banaszeketal2013. In the general case, such a setup has slit detectors
of varying efficiency, denoted by a parameter $q\in[0,1]$ that can be chosen
in the setup. The possible degrees of efficiency are of course mutually
exclusive, so these setups are mutually exclusive.

The slit detector has a given efficiency in the following sense:

Let us call $y$ the detection position on the screen, and $X_{1}$ is the
statement that detection occurs at slit #1 (you can translate to random-
variable jargon if you prefer). When we prepare the electromagnetic field in a
quantum state $S$, and use ideal detectors with perfect efficiency, the
probability of detection at slit #1 is, say $p_{S}$, and $1-p_{S}$ for slit
#2.

If we use the setup with detectors having efficiency $q$ – denote it by
$D_{q}$ – then the probability of detection at slit #1 is

$\mathrm{p}(X_{1}\nonscript\,|\nonscript\;\mathopen{}D_{q},S)=\tfrac{1}{2}(1-q)+q\,p_{S}\;,$
(11)

and $\tfrac{1}{2}(1+q)-q\,p_{S}$ for slit #2.

The setup with perfect detectors is the limit case $q=1$. In the case of zero
efficiency, $q=0$, there is no relation between the light states and the
firing of the slit detectors; that is, we are always fully uncertain as to
which detector would fire, no matter how the light state is prepared. These
kinds of setup – and many other interesting ones – are quite easy to prepare
with the statistically analogous quantum Mach-Zehnder-like interferometers
(see the textbooks in footnote 1 below; [§ 4.2]leonhardt1997yuenetal1978).

In each setup $D_{q}$ (and given the light state $S$) we also have the
conditional probability distribution
$\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}D_{q},S)$ for detection at
$y$ on the screen, and the conditional probability distributions
$\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}X,D_{q},S)$ for detection at
$y$ on the screen, given detection $X$ at the slits. We have

$\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}D_{q},S)={}\\\
\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}X_{1},D_{q},S)\;\mathrm{p}(X_{1}\nonscript\,|\nonscript\;\mathopen{}D_{q},S)+\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}X_{2},D_{q},S)\;\mathrm{p}(X_{2}\nonscript\,|\nonscript\;\mathopen{}D_{q},S)\;.$
(12)

This is an instance of the conditional-probability rule, which is of course
valid. This equality also holds for long-run frequencies (see point (iii)
below). Note that such conditional and unconditional frequencies are
experimentally observed. I would like you to convince yourself, though, that
the _equality_ above (not the specific values of the frequencies) is not
really an experimental fact, since it rests on the very way we measure
conditional frequencies.

The conditional and unconditional distributions above will of course be
different depending on the setup $D_{q}$ and the light state $S$. But in each
instance the rule of conditional probability holds. For example, if
${\color[rgb]{0.265625,0.46484375,0.66796875}q^{\prime}}\neq{\color[rgb]{0.66796875,0.19921875,0.46484375}q^{\prime\prime}}$,

$\displaystyle\begin{multlined}\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.265625,0.46484375,0.66796875}D_{q^{\prime}}},S)={}\\\
\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}X_{1},{\color[rgb]{0.265625,0.46484375,0.66796875}D_{q^{\prime}}},S)\;\mathrm{p}(X_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.265625,0.46484375,0.66796875}D_{q^{\prime}}},S)+\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}X_{2},{\color[rgb]{0.265625,0.46484375,0.66796875}D_{q^{\prime}}},S)\;\mathrm{p}(X_{2}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.265625,0.46484375,0.66796875}D_{q^{\prime}}},S)\;,\end{multlined}\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.265625,0.46484375,0.66796875}D_{q^{\prime}}},S)={}\\\
\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}X_{1},{\color[rgb]{0.265625,0.46484375,0.66796875}D_{q^{\prime}}},S)\;\mathrm{p}(X_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.265625,0.46484375,0.66796875}D_{q^{\prime}}},S)+\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}X_{2},{\color[rgb]{0.265625,0.46484375,0.66796875}D_{q^{\prime}}},S)\;\mathrm{p}(X_{2}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.265625,0.46484375,0.66796875}D_{q^{\prime}}},S)\;,$
(15)
$\displaystyle\begin{multlined}\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)={}\\\
\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}X_{1},{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)\;\mathrm{p}(X_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)+\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}X_{2},{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)\;\mathrm{p}(X_{2}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)\;,\end{multlined}\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)={}\\\
\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}X_{1},{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)\;\mathrm{p}(X_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)+\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}X_{2},{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)\;\mathrm{p}(X_{2}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)\;,$
(18)
$\displaystyle\begin{multlined}\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.265625,0.46484375,0.66796875}D_{q^{\prime}}},S)\mathrel{\bm{\neq}}{}\\\
\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}X_{1},{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)\;\mathrm{p}(X_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)+\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}X_{2},{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)\;\mathrm{p}(X_{2}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)\;.\end{multlined}\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.265625,0.46484375,0.66796875}D_{q^{\prime}}},S)\mathrel{\bm{\neq}}{}\\\
\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}X_{1},{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)\;\mathrm{p}(X_{1}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)+\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}X_{2},{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)\;\mathrm{p}(X_{2}\nonscript\,|\nonscript\;\mathopen{}{\color[rgb]{0.66796875,0.19921875,0.46484375}D_{q^{\prime\prime}}},S)\;.$
(21)

The last inequality, analogous to eq. (7), comes from experimental
observations (see the brief discussion below about the relation with de
Finetti’s theorem), and was not in fact not ruled out a priori by the
probability calculus.

Now let me discuss a couple of very interesting experimental facts about this
collection of setups:

First, _both the conditional $y|X$ and unconditional_ probability
distributions for the screen detection $y$ generally _have an oscillatory
profile, typical of interference_ [woottersetal1979, banaszeketal2013][see
also][for other experimental variations]chiaoetal1995. The oscillatory
character is maximal for the zero-efficiency setup $q=0$ and decreases as $q$
increases. For the perfect-detector setup $q=1$ there is no interference. But
we can have quite a lot of interference even when the detection efficiency is
quite high, so that for some light states we are almost certain about slit
detection; see references above. (The profile depends on the specific light
state, of course, which we are assuming fixed.)

Second, the unconditional (frequency) distribution observed in the setup
$D_{0}$ with zero-efficiency slit detectors is experimentally equal to the
distribution for screen detection observed in the setup without slit detectors
(note that in the latter setup we cannot speak of conditional or unconditional
probability, since slit detection does not exist).

Third, one conditional distribution observed in the setup with one slit closed
is experimentally equal to one in the setup $D_{1}$ with perfect slit
detectors. (Here we must be careful, because there is no slit detection in the
second setup; rather, we speak of appearance or non-appearance at the screen,
and in the latter case no conditional distribution is defined.)

The equalities in the last two cases should a priori not be expected, because
the setups are physically different. Of course one can look for physical,
“hidden variables” explanations of such equalities. Experimental quantum
optics simply acknowledges the fact that two setups are equivalent for such
detection purposes, and incorporates this information into its mathematical
formalism, by means of appropriately defined ‘ povms ’, discussed below.

Note the statistical analogy between the cases above and the cases with the
setups of the urn examples previously discussed. In each setup, the rule of
conditional probability holds (and in the quantum case we can have
distributions, conditional and unconditional, with oscillatory profiles).
Across different setups, probability theory says that such a rule cannot be
applied; and indeed we find inequalities across some setups and equalities
across others, both in the quantum and non-quantum case, eqs (7), (10). Even
more striking statistical analogies appear in the already cited non-quantum
counter-examples [kirkpatrick2001_r2003, kirkpatrick2002_r2003][§
IV]portamana2003_r2004.

It is also possible to consider situations in which we are uncertain about
which measurement setup applies. For example we may not know whether there
were slit detectors, or the value $q$ of the detector efficiency. In such
situations we introduce probabilities $\mathrm{p}(D_{\dotso})$ for the
possible setups and the conditional-probability rule applies, yielding for
example

$\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}S)=\mathop{\textstyle\sumop\slimits@}\nolimits_{q}\mathrm{p}(y\nonscript\,|\nonscript\;\mathopen{}D_{q},S)\;\mathrm{p}(D_{q})$
(22)

(here our knowledge of the state was assumed to be irrelevant to our inference
about the setup). Then, given the measurement outcome, we can make inferences
about the setup [barnettetal2003, zimanetal2004_r2006, darianoetal2004][see
also][]rigoetal1998 – for example whether a slit detector was present or not –
again using the conditional-probability rule in the guise of Bayes’s theorem.
This kind of inference is especially important in quantum key distribution
[nielsenetal2000_r2010], where we try to infer whether a third party was
eavesdropping, that is, performing a covert measurement. Again no violations
of the probability rules in the quantum realm: quite the opposite, those rules
allow us to make important inferences.

## Further remarks and curiosities about quantum interference experiments

I would like to mention a couple more experimental facts – which are, besides,
statistically very interesting – to correct some statements by Gelman & Yao in
relation to the double-slit experiment.

  1. (i)

It _does_ matter whether many photons are sent at once, or one at a time
[gelmanetal2020, cf.]; as well as their wavelength, temporal spread, and so on
(strictly speaking, the spatio-temporal dependence of the field mode). These
details are part of the specification of the light state $S$ mentioned above,
and lead to different probabilities distributions of screen detection.

For example, in some setups and for some states we can have a detection
probability density $\mathrm{p}(y_{1})$ for the first photon, and a
_different_ density for the second photon
$\mathrm{p}(y_{2}\nonscript\,|\nonscript\;\mathopen{}y_{1})$, conditional on
the detection of the first – both being different from the cumulative density
of detections. Interference phenomena can also be observed in time, not only
in space. See e.g. the phenomena of higher-order coherence, bunching, anti-
bunching, and many other interesting ones111[for
example][]mandeletal1965,morganetal1966,paul1982,jacobsonetal1995[and
textbooks such
as][]loudon1973_r2000,mandeletal1995_r2008,scullyetal1997_r2001,gerryetal2005,wallsetal1994..
Quoting Glauber *[Lect. I p. 65]glauber1965:

> The new light detectors enable us to ask more subtle questions than just
> ones about average intensities; we can for example, ask questions about the
> counting of pairs of quanta, and can make measurements of the probability
> that the quanta are present at an arbitrary pair of space points, at an
> arbitrary pair of limes.

The rules of the probability calculus also apply in all such situations. We
can infer, for example, the position of the first photon detection given the
second from the conditional probability rule
$\mathrm{p}(y_{1}\nonscript\,|\nonscript\;\mathopen{}y_{2})\propto\mathrm{p}(y_{2}\nonscript\,|\nonscript\;\mathopen{}y_{1})\;\mathrm{p}(y_{1})$.

  2. (ii)

The details about the light source and the setup are not “latent variables”:
they specify the quantum state of light and the measurement performed on it.
They are like the initial and boundary conditions necessary for the
specification of the behaviour of any physical system.

  3. (iii)

In view of point (i) above, it is important not to conflate the probability
distributions for single-photon detections, those for cumulative photon
detection, and the _frequency_ distributions of a long-run of such detections
[gelmanetal2020, § 2, seem to conflate the two]. Such distinction is always
important from a Bayesian point of view. Quoting Glauber *[Lect. I p.
70]glauber1965 again:

> There is therefore no possibility _in general_ of replacing the ensemble
> averages by time averages. […] we shall find that individual measurements
> yield results wholly unlike their ensemble averages. The distinction between
> particular measurements and their averages may thus be quite essential.

I may add that the idea and parlance of “photons passing through slits” are
used today only out of tradition; maybe a little poetically. The technical
parlance, as routinely used in quantum-optics labs for example [leonhardt1997,
gerryetal2005], has a different underlying picture. The ‘ system ’ in a
quantum-optics experiment is not photons, but the modes of the field-
configuration operator11footnotemark: 1 (note that this is not yet Quantum
ElectroDynamics). “Photon numbers” denote the discrete outcomes of a specific
energy-measurement operator; “photon states” denote specific states of the
field operators. As another example, “entanglement” is strictly speaking not
among photons, but among modes of the field operator [vanenk2003b]. Several
quantum physicists indeed oppose the idea and parlance of “photons”, owing to
the confusion they lead to. Lamb [[, of the Lamb shift,]]lambetal1947 wrote in
[*]lamb1995:

> the author does not like the use of the word “photon”, which dates from
> 1926. In his view, there is no such thing as a photon. Only a comedy of
> errors and historical accidents led to its popularity among physicists and
> optical scientists.

Wald *wald1994 warns:

> standard treatments of quantum field theory in flat spacetime rely heavily
> on Poincaré symmetry (usually entering the analysis implicitly via plane-
> wave expansions) and interpret the theory primarily in terms of a notion of
> “particles”. Neither Poincaré (or other) symmetry nor a useful notion of
> “particles” exists in a general, curved spacetime, so a number of the
> familiar tools and concepts of field theory must be “unlearned” in order to
> have a clear grasp of quantum field theory in curved spacetime. [p. ix] […]
> the notion of “particles” plays no fundamental role either in the
> formulation or interpretation of the theory. [p. 2]

See also Davies’s _Particles do not exist_ *davies1984.

## A summary of the modern formalism of quantum theory

It may be useful to give a summary of how probability enters the modern
formalism of quantum theory. See textbooks such as Holevo *holevo1980_t2011,
Busch et al. *buschetal1995b, Peres *[especially ch. 12]peres1993_r2002, de
Muynck *[especially ch. 3]demuynck2002b_2004, and the excellent text by
Bengtsson & Życzkowski *bengtssonetal2006_r2017.

A quantum system is defined by its sets of possible states and possible
measurements. A state $\rho$ is represented by an Hermitean, positive-
definite, unit-trace matrix $\bm{\rho}$ [jakobczyketal2001, kimura2003,
kimuraetal2004, bengtssonetal2006_r2017, which satisfies additional
mathematical properties:], called ‘ density matrix ’. States traditionally
represented by kets $\lvert\psi\rangle$ are just special cases of density
matrices. A measurement setup $M$ is represented by a set of Hermitean,
positive-definite matrices $\\{\textbf{{{M}}}_{\bm{r}}\\}$ (of the same order
as the density matrices) adding up to the identity matrix. They are called ‘
positive-operator-valued measures ’, usually abbreviated povms. Traditional
von Neumann projection operators
$\\{\lvert\phi_{r}\rangle\langle\phi_{r}\rvert\\}$ are just special cases of
povms. Each matrix $\textbf{{{M}}}_{\bm{r}}$ is associated with an outcome
$\bm{r}$ of the measurement. These outcomes are mutually exclusive. An outcome
can actually represent a combination of simpler outcomes,
$\bm{r}\equiv(x,y,z,\dotsc)$, such as the intensities or firings at two or
more detectors.

The probability of observing outcome $\bm{r}\equiv(x,y,\dotsc)$ given the
measurement setup $M$ and the state $S$ is encoded in the trace-product of the
respective matrices:

$\mathrm{p}(x,y,\dotsc\nonscript\,|\nonscript\;\mathopen{}M\land
S)\equiv\operatorname{tr}(\textbf{{{M}}}_{x,y,\dotsc}\;\bm{\rho})\;,$ (23)

These probabilities for all $\bm{r}$ form a probability distribution. The
traditional Born-rule expression ‘
$\lvert\langle\phi_{r}|\psi\rangle\rvert^{2}$ ’ is just a special case of the
above formula. The probabilities in the formula come from repeated measurement
observations in the same experimental conditions: we can invoke de Finetti’s
*definetti1937,definetti1938 theorem here – the partial-exchangeability
variant – and some quantum physicists indeed do [cavesetal2002,
vanenketal2002, fuchsetal2004b]. The trace-product above is just a scalar
product in a particular space. How a set of probability or frequency
distributions can be encoded in scalar products is explained in a down-to-
earth way in Porta Mana *portamana2003b,portamana2004b.

Once the probability distribution above is given we can use the full-fledged
probability calculus for our inferences. We can for example sum (or integrate)
over detector outcomes $y,\dotsc{}$, obtaining the marginal probability for
detector outcome $x$; or calculate the probability of outcome $y$ conditional
on $x$; or make inferences about the measurement setup or the state. Again,
there are no violations of the probability rules. The formalism (23) is neat
in this respect because it allows us to represent such situations through new
povms and density matrices. You can easily check, for example, that the
marginal probability for $x$ from eq. (23) can be encoded in the povm
$\\{\textbf{{{M}}}^{\prime}_{x}\\}\equiv\\{\mathop{\textstyle\sumop\slimits@}\nolimits_{y,\dotsc}\textbf{{{M}}}_{x,y,\dotsc}\\}$.
A situation of uncertainty between setups $M^{\prime}$ and $M^{\prime\prime}$,
as in eq. (22), can be encoded in the povm
$\big{\\{}\mathrm{p}(M^{\prime})\;\textbf{{{M}}}^{\prime}_{\bm{r}}+\mathrm{p}(M^{\prime\prime})\;\textbf{{{M}}}^{\prime\prime}_{\bm{r}}\big{\\}}$.
And so on, and similarly for states and their density operators.

For systems with infinite degrees of freedom such as electromagnetic fields or
electrons (Fermionic fields), the matrices above are replaced by operators
defined in particular algebras. A povm element can actually be a space-time-
indexed operator. The computational details can become quite complicated, but
the same basic ideas apply.

This formalism obviously also includes the specification of post-measurement
states (if the system still exists afterwards), transformations, evolutions. I
shall not discuss these; see the textbooks cited above.

## Conclusions

I hope that the above discussion and bibliography clearly show that:

  * •

the rules of probability theory, including the conditional-probability rule,
are fully valid in quantum theory and essential in its modern applications;

  * •

some peculiar equalities or inequalities across different experimental
conditions do not contradict the conditional-probability rule, and they appear
just as well in quantum as in non-quantum situations, such as drawing from an
urn.

Quantum theory already has its physically conceptual difficulties and
computational difficulties, as should be clear from the portrait sketched in
the present comment. It is pointless – and pedagogically confusing and
detrimental, for students of quantum optics for instance – to make it seem
even more difficult with false claims of non-validity of probability theory or
with distorted pictures of its experimental content.

prenote(“de $X$” is listed under D, “van $X$” under V, and so on, regardless
of national conventions.)


# Brezis–Kato Type Regularity Results for Higher Order Elliptic Operators

Jakub Siemianowski
Institute of Mathematics,
Polish Academy of Sciences,
ul. Śniadeckich 8, 00-656 Warsaw, Poland
and
Faculty of Mathematics and Computer Sciences,
Nicolaus Copernicus University in Toruń
ul. Gagarina 11, 87-100 Toruń, Poland
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We prove a Brezis–Kato regularity type results for solutions of the higher
order nonlinear elliptic equation

$Lu=g(x,u)\qquad\text{in }\Omega$

with an elliptic operator $L$ of $2m$ order with variable coefficients and a
Carathéodory function $g:\Omega\times\mathbb{C}\to\mathbb{C}$, where
$\Omega\subset\mathbb{R}^{N}$ is an open set with $N>2m$.

###### Key words and phrases:

Brezis–Kato theorem, higher order elliptic operators, elliptic regularity,
polyharmonic operators

###### 2010 Mathematics Subject Classification:

35B65, 35J30, 35J61, 35G20

##  1\. Notations and Intorduction

The aim of this paper is to generalize the Brezis–Kato theorem [Brezis_Kato]
for higher order elliptic differential operators with variable coefficients.
It is intended to make the least assumptions possible concerning the
coefficients of the operator so that it can serve for future references.

We consider a linear differential operator $L=(-1)^{m}\sum_{|\alpha|\leq
2m}a_{\alpha}(x)D^{\alpha}$ with variable complex-valued coefficients defined
in an arbitrary open subset $\Omega$ of $\mathbb{R}^{N}$. Here for every
multi-index $\alpha=(\alpha_{1},\ldots,\alpha_{N})$ we set
$D^{\alpha}=\frac{\partial^{|\alpha|}}{\partial
x_{1}^{\alpha_{1}}\ldots\partial x_{N}^{\alpha_{N}}}$ and
$\xi^{\alpha}=\xi_{1}^{\alpha_{1}}\ldots\xi_{N}^{\alpha_{N}}$, for a vector
$\xi=(\xi_{1},\ldots,\xi_{N})$ . The characteristic form $l(x,\xi)$ of $L$ is
defined by $l(x,\xi)=\sum_{|\alpha|=2m}a_{\alpha}(x)\xi^{\alpha}$ for
$x\in\Omega$ and real vectors $\xi$. We say that $L$ is _strongly elliptic_ if
there exists $\lambda>0$ such that

(1.1) $\mathfrak{Re}(l(x,\xi))\geq\lambda|\xi|^{2m}$

for $x\in\Omega$ and all real vectors $\xi=(\xi_{1},\ldots,\xi_{N})$. We make
standing assumptions:

  1. (i)

all coefficients $a_{\alpha}$ belong to the space
$L^{\infty}_{\mathrm{loc}}(\Omega)$,

  2. (ii)

top-order coefficients $a_{\alpha}$ for $|\alpha|=2m$ are continuous on
$\Omega$.

We denote by $C^{k}(\Omega)$ (resp. $C^{\infty}_{0}(\Omega)$) the class of
complex valued functions defined in $\Omega$ which has continuous derivatives
up to order $k$ (resp. which are smooth and have compact support). We consider
Hölder’s spaces $C^{k,h}(\Omega)$ for nonnegative integers $k$ and $0<h<1$
which consists of all functions $u\in C^{k}(\Omega)$ such that $D^{\alpha}u$
with $|\alpha|=k$ satisfies the (uniform) Hölder condition of exponent $h$.
The symbol $W^{k,p}(\Omega)$ for $1\leq p<\infty$ denotes the Sobolev space of
all complex valued $u\in L^{p}(\Omega)$ for which all the distributional
derivatives $D^{\alpha}u$ are also in $L^{p}(\Omega)$ for $|\alpha|\leq k$. We
consider the norm $\|u\|_{W^{k,p}(\Omega)}=\left(\sum_{|\alpha|\leq
k}\|D^{\alpha}u\|^{p}_{L^{p}(\Omega)}\right)^{1/p}$ for $u\in
W^{k,p}(\Omega)$. We use the symbol $W^{k,p}_{0}(B)$ to denote the closure of
$C^{\infty}_{0}(\Omega)$ in the $W^{k,p}(\Omega)$ norm. The local Sobolev
space $W^{k,p}_{\mathrm{loc}}(\Omega)$ consists of functions
$u:\Omega\to\mathbb{C}$ such that $u\in W^{k,p}(U)$ for every open subset
$U\subset\subset\Omega$ (the latter means that $\overline{U}$ is compact and
$\overline{U}\subset\Omega$). Similarly, $C^{k,h}_{\mathrm{loc}}(\Omega)$
consists of $u\in C^{k}(\Omega)$ such that $u\in C^{k,h}(U)$ for every open
$U\subset\subset\Omega$. We see with the aid of the partition of unity that a
function $u:\Omega\to\mathbb{C}$ belongs to $W^{k,p}_{\mathrm{loc}}(\Omega)$
if and only if $u\in W^{k,p}(B)$ for every ball $B\subset\subset\Omega$.

We use $\lesssim$ to denote the inequality up to a constant whenever it is
convenient. Throughout what follows, $g:\Omega\times\mathbb{C}\to\mathbb{C}$
stands for a Carathéodory function, i.e., measurable with respect to the first
variable and continuous with respect to the second one.

The Brezis–Kato theorem — Thm 2.3 in [Brezis_Kato] — asserts that if $u\in
W^{1,2}_{\mathrm{loc}}(\Omega)$ is a weak solution of

$-\Delta u=g(x,u)\ \text{in }\Omega,$

that is for every $\varphi\in C^{\infty}_{0}(\Omega)$

$\int_{\Omega}\nabla u\nabla\varphi\,dx=\int_{\Omega}g(x,u)\varphi\,dx$

and there exists a non-negative $a\in L^{\frac{N}{2}}_{\mathrm{loc}}(\Omega)$
such that

$|g(x,u(x))|\leq a(x)(1+|u(x)|)\ \text{a.e. in }\Omega,$

then $u\in\bigcap_{1\leq q<\infty}L^{q}_{\mathrm{loc}}(\Omega)$. See also
Lemma B.3 in Struwe’s book [Struwe] for a direct exposition of this result.

Notice that the regularity of solutions which the Brezis–Kato theorem asserts
is higher than the one obtained through the elliptic regularity theory. This
has numerous important consequences, let us name just two. The Brezis–Kato
theorem has proved to be useful in the study of elliptic equations through the
variational approach. It enables one to show that critical points of certain
functionals are classical $C^{2}$-solutions, see [Struwe] and [Willem]. For
another example, the Brezis–Kato theorem turns out to be crucial in proving
the so-called Pohozaev identity, cf. [Berestycki_Lions].

The Brezis–Kato theorem has been extended to the case of the nonlocal right
hand side in [Moroz] and generalized to the case of the fractional Laplacian
in [Leite]. The generalization to the bilaplacian operator has recently been
proved in [MS]. To the best of the author’s knowledge these are the only
results present in the literature.

In this paper we generalize the Brezis–Kato theorem in two directions.
Firstly, we consider the general elliptic differential operator with variable
coefficients. Secondly, we allow the elliptic operator to be of any even
order. Results presented here seems to be new even for the second order
differential operator.

The proof of the main result Theorem 2.1 is inspired by [Van_der_Vorst]. It
was developed and adjusted to the bilaplacian case in [MS] and is presented
here in a full generality.

The organization of the paper is the following. In Section 2 we provide the
proof of the main result — Theorem 2.1 — divided into some number of lemmata.
Section 3 contains Theorem 3.2 and Theorem 3.3 which are just versions of
Theorem 2.1 adjusted to the weaker notions of solutions occurring in the
literature.

##  2\. The Main Result

######  Theorem 2.1.

Let $m\geq 1$, $N>2m$ be positive integers, $p>1$ and
$\Omega\subset\mathbb{R}^{N}$ be open. Let $u\in
W^{2m,p}_{\mathrm{loc}}(\Omega)$ be a _strong solution_ of $Lu=g(x,u)$, i.e.,

(2.1) $Lu=g(x,u(x))\quad\text{a.e. in }\Omega,$

If there exists a non-negative $a\in L^{\frac{N}{2m}}_{\mathrm{loc}}(\Omega)$
such that

(2.2) $\displaystyle|g(x,u(x))|\leq a(x)(1+|u(x)|),\quad\text{a.e. in
}\Omega,$

then $u\in\bigcap_{1\leq q<\infty}L^{q}_{\mathrm{loc}}(\Omega)$. If $g$ has a
polynomial rate of growth, namely, there exists $r\geq 1$ such that

(2.3) $\displaystyle|g(x,s)|\lesssim 1+|s|^{r}\quad\text{for all
}x\in\Omega\text{ and }s\in\mathbb{C},\;\text{where
}r=\begin{cases}\frac{N}{N-2mp}&\text{if }p<\frac{N}{2m},\\\ \text{ is
arbitary}&\text{if }p\geq\frac{N}{2m},\end{cases}$

then $u\in W^{2m,q}_{\mathrm{loc}}\cap C^{2m-1,h}_{\mathrm{loc}}(\Omega)$ for
all $1\leq q<\infty$ and $0<h<1$.

######  Remark 2.2.

Observe that the hypothesis $N>2m$ in Theorem 2.1 is the only interesting
case, for otherwise the Sobolev embedding
$W^{2m,1}_{\mathrm{loc}}(\Omega)\subset\bigcap_{1\leq
q<\infty}L^{q}_{\mathrm{loc}}(\Omega)$ immediately implies the assertion for
any strong solution. Form now on we confine our attention to $N>2m$ without
further comment.

Let us also emphasize that without stronger assumptions Theorem 2.1 is false
in general for $p=1$ , as there is a counterexample, see Theorem 3 in
[escauriaza].

Before proving Theorem 2.1, we need some preparations first.

######  Proposition 2.3.

Let $s\geq\frac{N}{N-2m}$. If for some $u\in L^{s}_{\mathrm{loc}}(\Omega)$
there is a non-negative $a\in L^{\frac{N}{2m}}_{\mathrm{loc}}(\Omega)$ such
that

$|g(x,u(x))|\leq a(x)(1+|u(x)|),\quad\text{a.e. in }\Omega,$

then $g(\cdot,u)\in L^{\frac{Ns}{N+2ms}}_{\mathrm{loc}}(\Omega)$.

###### Proof.

For any ball $B\subset\subset\Omega$, we have

$\int_{B}|g(x,u)|^{\frac{Ns}{N+2ms}}\,dx\lesssim\int_{B}a^{\frac{Ns}{N+2ms}}\,dx+\int_{B}\left(a|u|\right)^{\frac{Ns}{N+2ms}}\,dx.$

Since $\frac{Ns}{N+2ms}<\frac{N}{2m}$, the first integral
$\int_{B}a^{\frac{Ns}{N+2ms}}\,dx$ is finite. We use the Hölder inequality
with the exponents

$\frac{1}{\frac{N+2ms}{2ms}}+\frac{1}{\frac{N+2ms}{N}}=1$

to get

$\int_{B}\left(a|u|\right)^{\frac{Ns}{N+2ms}}\,dx\leq\left(\int_{B}a^{\frac{N}{2m}}\,dx\right)^{\frac{2ms}{N+2ms}}\left(\int_{B}|u|^{s}\,dx\right)^{\frac{N}{N+2ms}}<\infty.$

Therefore $\int_{B}|g(x,u)|^{\frac{Ns}{N+2ms}}\,dx$ is finite what completes
the proof. ∎

The following lemma is based on Lemma B.2 from [Van_der_Vorst].

######  Lemma 2.4.

Let $1\leq p<\infty$ and $U\subset\mathbb{R}^{N}$ be a measurable set with
$0<|U|<\infty$, where $|U|$ denotes the measure of $U$. We assume that $a\in
L^{p}(U)$ and $v\in L^{1}(U)$. For every $\varepsilon>0$ there exists
$q_{\varepsilon}\in L^{p}(U)$ and $f_{\varepsilon}\in L^{\infty}(U)$ such that
$\|q_{\varepsilon}\|_{L^{p}(U)}<\varepsilon$ and

$a(x)v(x)=q_{\varepsilon}(x)v(x)+f_{\varepsilon}(x)\quad\text{for a.e. }x\in
U.$

###### Proof.

We define the sets

$A_{k}=\left\\{x\in U\mid|a(x)|<k\right\\}\quad\text{and}\quad
B_{k}=\left\\{x\in U\mid|v(x)|<k\right\\}.$

Since $a$, $v\in L^{1}(U)$, we have $|A_{k}^{c}|$, $|B_{k}^{c}|\to 0$. Observe
that

$\displaystyle|A_{k}^{c}\cup B_{k}^{c}|=|A_{k}^{c}|+|B_{k}^{c}|-|A_{k}^{c}\cap
B_{k}^{c}|\leq|A_{k}^{c}|+|B_{k}^{c}|\to 0,$

so there is $k_{1}$ such that for $k\geq k_{1}$

$|A_{k}\cap B_{k}|=|U|-|A_{k}^{c}\cup B_{k}^{c}|>0.$

Fix $\varepsilon>0$. By the absolute continuity of the measure
$\mu(C):=\int_{C}|a(x)|^{p}\,dx$, we find $k_{2}\geq k_{1}$ such that

(2.4) $\int_{A_{k}^{c}\cup
B_{k}^{c}}|a(x)|^{p}\,dx<\left(\frac{1}{2}\varepsilon\right)^{p},\quad\text{for
}k\geq k_{2}.$

We are in a position to define

$\displaystyle q_{\varepsilon}(x):$
$\displaystyle=\begin{cases}\frac{1}{k}a(x),&\text{for }x\in A_{k}\cap
B_{k},\\\ a(x),&\text{for }x\in(A_{k}\cap
B_{k})^{c},\end{cases}\quad\text{and}\quad f_{\varepsilon}$
$\displaystyle:=(a(x)-q_{\varepsilon}(x))v(x).$

We estimate using the definition of $q_{\varepsilon}$ and (2.4), for $k\geq
k_{2}$,

$\displaystyle\|q_{\varepsilon}\|_{L^{p}(U)}$
$\displaystyle\leq\frac{1}{k}\left(\int_{A_{k}\cap
B_{k}}|a(x)|^{p}\,dx\right)^{1/p}+\left(\int_{(A_{k}\cap
B_{k})^{c}}|a(x)|^{p}\right)^{1/p}\,dx<\frac{1}{k}\|a\|_{L^{p}(U)}+\frac{1}{2}\varepsilon.$

If $k\geq\max\left\\{\frac{2\|a\|_{L^{p}(U)}}{\varepsilon},k_{2}\right\\}$,
then

$\|q_{\varepsilon}\|_{L^{p}(U)}<\varepsilon.$

Observe that $f_{\varepsilon}=0$ on $(A_{k}\cap B_{k})^{c}$, hence we get

$|f_{\varepsilon}(x)|=\left|a(x)-\frac{1}{k}a(x)\right||v(x)|\mathds{1}_{A_{k}\cap
B_{k}}(x)=\frac{k-1}{k}|a(x)||v(x)|\mathds{1}_{A_{k}\cap B_{k}}(x)\leq(k-1)k,$

in view of the definitions of $A_{k}$ and $B_{k}$. ∎

The next lemma is crucial. It is a modification of Lemma 2.2 in [MS].

######  Lemma 2.5.

If $u\in W^{2m,p}_{\mathrm{loc}}(\Omega)$, with $p>1$, is a strong solution of

$Lu=g(x,u)\quad\text{in }\Omega,$

and (2.2) holds, then

$u\in\begin{cases}L^{\frac{Np}{N-(2m+1)p}}_{\mathrm{loc}}(\Omega),&\text{if
}p<\frac{N}{2m+1},\\\ \bigcap_{1\leq
q<\infty}L^{q}_{\mathrm{loc}}(\Omega),&\text{if
}p\geq\frac{N}{2m+1}.\end{cases}$

###### Proof.

Observe that if $2mp\geq N$, then we are done due to the Sobolev embedding
$W^{2m,p}_{\text{loc}}(\Omega)\subset L^{q}_{\text{loc}}(\Omega)$ for $1\leq
q<\infty$. We therefore assume that $2mp<N$.

Take some ball $B_{0}\subset\subset\Omega$ and choose another ball $B$ such
that $B_{0}\subset\subset B\subset\subset\Omega$. Take a smooth cut-off
function $\eta\in C^{\infty}_{0}(B)$ satisfying $\eta\equiv 1$ on $B_{0}$ and
$0\leq\eta\leq 1$. Let us define

$\displaystyle\tilde{a}(x)$
$\displaystyle:=\frac{g(x,u(x))}{u(x)}\mathds{1}_{\left\\{x\mid|u(x)|>1\right\\}}(x),$
$\displaystyle b(x)$
$\displaystyle:=g(x,u(x))\mathds{1}_{\left\\{x\mid|u(x)|\leq 1\right\\}}(x),$

for $x\in B$, where it is understood that $\tilde{a}(x)=0$ whenever $u(x)=0$.
We then have

$g(x,u(x))=\tilde{a}(x)u(x)+b(x),\quad x\in B,$

and by (2.2)

$\displaystyle|\tilde{a}(x)|$
$\displaystyle\leq\frac{a(x)(1+|u(x)|)}{|u(x)|}\mathds{1}_{\left\\{x\mid|u(x)|>1\right\\}}(x)\leq\frac{a(x)(2|u(x)|)}{|u(x)|}\mathds{1}_{\left\\{x\mid|u(x)|>1\right\\}}(x)\leq
2a(x),$ $\displaystyle|b(x)|$ $\displaystyle\leq
a(x)(1+|u(x)|)\mathds{1}_{\left\\{x\mid|u(x)|\leq 1\right\\}}(x)\leq 2a(x),$

so $\tilde{a}$, $b\in L^{\frac{N}{2m}}(B)$. Moreover, $u$ is a strong solution
of

(2.5) $Lu=\tilde{a}(x)u+b(x)\quad\text{in }B.$

Let us consider

$v:=u\eta\in W^{2m,p}(B)\cap W^{m,p}_{0}(B).$

By the Leibniz formula for any multi-index $|\alpha|\leq 2m$ we get

$D^{\alpha}v=\left(D^{\alpha}u\right)\eta+\sum_{\beta<\alpha}\binom{\alpha}{\beta}D^{\beta}uD^{\alpha-\beta}\eta,$

and so

(2.6) $\displaystyle Lv$ $\displaystyle=(-1)^{m}\sum_{|\alpha|\leq
2m}a_{\alpha}(x)D^{\alpha}v=\sum_{|\alpha|\leq
2m}a_{\alpha}(x)\left(\left(D^{\alpha}u\right)\eta+\sum_{\beta<\alpha}\binom{\alpha}{\beta}D^{\beta}uD^{\alpha-\beta}\eta\right)$
$\displaystyle=\left(Lu\right)\eta+F(u,\eta),$

where

$F(u,\eta)=\sum_{\beta<\alpha,\,|\alpha|=2m}c_{\alpha,\beta}(x)D^{\beta}uD^{\alpha-\beta}\eta,$

for some coefficients $c_{\alpha,\beta}\in L^{\infty}(B)$. Since $u\in
W^{2m,p}(B)\subset W^{2m-1,\frac{Np}{N-p}}(B)$, we estimate

(2.7)
$\left\|F(u,\eta)\right\|_{L^{\frac{Np}{N-p}}(B)}\lesssim\|u\|_{W^{2m-1,\frac{pN}{N-p}}(B)}\|\eta\|_{W^{2m,\infty}(B)}\lesssim\|u\|_{W^{2m,p}(B)}.$

By Lemma 2.4, for every $\varepsilon>0$ there is $q_{\varepsilon}\in
L^{\frac{N}{2m}}(B)$ and $\widehat{f}_{\varepsilon}\in L^{\infty}(B)$ such
that

(2.8)
$\tilde{a}(x)v(x)=q_{\varepsilon}(x)v(x)+\widehat{f}_{\varepsilon}(x)\quad\text{and}\quad\|q_{\varepsilon}\|_{L^{\frac{N}{2m}}(B)}<\varepsilon.$

We use (2.6), (2.5) and (2.8)

$\displaystyle Lv$
$\displaystyle=\left(\tilde{a}(x)u+b(x)\right)\eta+F(u,\eta)=\tilde{a}(x)v+b(x)\eta+F(u,\eta)$
$\displaystyle=q_{\varepsilon}(x)v+\underbrace{\widehat{f}_{\varepsilon}(x)+b(x)\eta}_{f_{\varepsilon}}+F(u,\eta)=q_{\varepsilon}(x)v+f_{\varepsilon}(x)+F(u,\eta),$

with

(2.9) $f_{\varepsilon}\in L^{\frac{N}{2m}}(B).$

We therefore arrive at

(2.10) $Lv-q_{\varepsilon}v=f_{\varepsilon}+F(u,\eta).$

######  Proposition 2.6.

Let $1<p<q<\frac{N}{2m}$ and $v\in W^{2m,p}(B)\cap W^{m,p}_{0}(B)$. If for
every $\varepsilon>0$ we have $Lv-q_{\varepsilon}v\in L^{q}(B)$ whith
$\|q_{\varepsilon}\|_{L^{\frac{N}{2m}}(B)}<\varepsilon$, then $v\in
W^{2m,q}(B)$.

###### Proof.

We recall some theory about differential and unbounded operators, see
[Browder] and [Kato] for details and proofs. Fix $1<s<\infty$. Let $A_{s}$ be
the $L^{s}(B)$–realization of the elliptic differential operator
$L=(-1)^{m}\sum_{|\alpha|\leq 2m}a_{\alpha}(x)D^{\alpha}$ satisfying (i) and
(ii) under null Dirichlet boundary conditions. The domain $D_{s}$ of $A_{s}$
equals to $W^{2m,s}(B)\cap W^{m,s}_{0}(B)$. For $u\in D_{s}$ we set
$A_{s}u=Lu=(-1)^{m}\sum_{|\alpha|\leq 2m}a_{\alpha}(x)D^{\alpha}u$ and the
latter is a well-defined element of $L^{s}(B)$. There exists a constant
$\tilde{k}_{s}>0$ such that for every $u\in D_{s}$

(2.11)
$\|u\|_{W^{2m,s}(B)}\leq\tilde{k}_{s}\left(\|Lu\|_{L^{s}(B)}+\|u\|_{L^{s}(B)}\right),$

see formula (1) in [Browder].

By Thm 2 of [Browder], the inverse operator $(A_{s}+\zeta I)^{-1}:L^{s}(B)\to
D_{s}\subset L^{s}(B)$ exists (notice that $D(A_{s}+\zeta I)=D_{s}$) for large
positive $\zeta$. We fix such a large $\zeta>0$ and consider the invertible
operator $\tilde{A}_{s}:=A_{s}+\zeta I$. For $f\in L^{s}(B)$ we set
$u=\tilde{A}_{s}^{-1}f\in D_{s}$. Then

$\|u\|_{L^{s}(B)}\leq\|\tilde{A}_{s}^{-1}\|\|f\|_{L^{s}(B)}=c_{s}\|\tilde{A}_{s}u\|_{L^{s}(B)}$

The above and the inequality (2.11) applied to the operator $L+\zeta I$ yields

(2.12) $\|u\|_{W^{2m,s}(B)}\leq k_{s}\|\tilde{A}_{s}u\|_{L^{s}(B)},\quad u\in
D_{s}.$

Let us also consider for $1<s<\frac{N}{2m}$ the multiplication operator
$Q_{\varepsilon,s}$ considered as an unbounded operator from $L^{s}(B)$ into
$L^{s}(B)$ defined by

$Q_{\varepsilon,s}u=q_{\varepsilon}u$

for $u$ in the domain $D(Q_{\varepsilon,s})$. Observe that for any $u\in
L^{\frac{Ns}{N-2ms}}(B)$ we have

(2.13)
$\|Q_{\varepsilon,s}\|_{L^{s}(B)}\leq\|q_{\varepsilon}\|_{L^{\frac{N}{2m}}(B)}\|u\|_{L^{\frac{Ns}{N-2ms}}(B)},$

where we used the Hölder inequality with the exponents

$\frac{1}{s}=\frac{1}{\frac{N}{2m}}+\frac{1}{\frac{Ns}{N-2ms}}.$

For $u\in D_{s}$ we utilize (2.13), the Sobolev embedding $W^{2m,s}(B)\subset
L^{\frac{Ns}{N-2ms}}(B)$ and (2.12) to deduce

$\|Q_{\varepsilon,q}u\|_{L^{q}(B)}\leq
c_{\mathrm{Sobolev}}\|q_{\varepsilon}\|_{L^{\frac{N}{2m}}(B)}\|u\|_{W^{2m,q}(B)}\leq
c_{\mathrm{Sobolev}}k_{s}\|q_{\varepsilon}\|_{L^{\frac{N}{2m}}(B)}\|\tilde{A}_{s}u\|_{L^{q}(B)}.$

Therefore $Q_{\varepsilon,s}$ is relatively bounded with respect to
$\tilde{A}_{s}$, see §4 of [Kato] for the definitions and details. If
$\|q_{\varepsilon}\|_{L^{\frac{N}{2m}}(B)}$ is small enough so that
$\|q_{\varepsilon}\|_{L^{\frac{N}{2m}}(B)}c_{\mathrm{Sobolev}}k_{s}<1$, then
$(\tilde{A}_{s}-Q_{\varepsilon,s})^{-1}:L^{s}(B)\to D_{s}\subset L^{s}(B)$
exists in view of Thm 1.16 in §4.4 of [Kato]. By (2.8) we can always assume
this is the case.

Case I: $q\leq\frac{Np}{N-2mp}$

In this case $v\in W^{2m,p}(B)\subset L^{\frac{Np}{N-2mp}}(B)\subset L^{q}(B)$
and so

$\underbrace{Lv-q_{\varepsilon}v}_{\in L^{q}(B)}+\underbrace{\zeta v}_{\in
L^{q}(B)}=:f\in L^{q}(B),$

where $\zeta$ is large enough so that both $\tilde{A}_{q}^{-1}$ and
$\tilde{A}_{p}^{-1}$ exist. We define

$y:=(\tilde{A}_{q}-Q_{\varepsilon,q})^{-1}f\in D_{q},$

then

$(\tilde{A}_{q}-Q_{\varepsilon,q})y=f.$

On the other hand, we notice that $v\in D_{p}$ so
$(\tilde{A}_{p}-Q_{\varepsilon,p})v$ is well-defined and equals to $f$. Since
$q>p$ and $D_{q}\subset D_{p}$, we may consider $\tilde{A}_{q}$ as a
restriction of $\tilde{A}_{p}$ and thus

$(\tilde{A}_{q}-Q_{\varepsilon,q})y=(\tilde{A}_{p}-Q_{\varepsilon,p})y.$

This yields

$(\tilde{A}_{p}-Q_{\varepsilon,p})(y-v)=0.$

Since $\tilde{A}_{p}-Q_{\varepsilon,p}$ is invertible, we get

$v=y\in D^{q}\subset W^{2m,q}(B)$

as claimed.

Case II: $q>\frac{Np}{N-2mp}$

We proceed similarly as in the previous case, but this time

$Lv-q_{\varepsilon}v+\zeta v\in L^{\frac{Np}{N-2mp}}(B),\quad\zeta>0,$

and, reasoning as above, we can conclude that $v\in D_{\frac{Np}{N-2mp}}$. We
define $p_{i}:=\frac{Np}{N-2mip}$ for $i=0,\ldots,k+1$, where $k\geq 1$ is the
integer satisfying

$k<\frac{N(q-p)}{2mpq}\leq k+1.$

Repeating this procedure $k$ times we see that $v\in D_{p_{k}}$ where $p_{k}$
satisfies $p_{k}<q\leq p_{k+1}$, and so $v\in W^{2m,p_{k}}(B)\subset
L^{p_{k+1}}(B)\subset L^{q}(B)$. We then have

$Lv-q_{\varepsilon}v+\zeta v\in L^{q}(B),\quad\zeta>0,$

and proceeding as in Case I we show that $u\in W^{2m,q}(B)$. ∎

We continue the proof of Lemma 2.5 by considering two cases. If
$p<\frac{N}{2m+1}$, then for $q=\frac{Np}{N-p}$ we estimate using (2.7)

$\|f_{\varepsilon}+F(u,\eta)\|_{L^{q}(B)}\lesssim\|f_{\varepsilon}\|_{L^{\frac{N}{2m}}(B)}+\|u\|_{W^{2m,p}(B)},$

By (2.9) and the hypothesis the $L^{q}(B)$ norm of the right hand side of
(2.10) is finite. By Proposition 2.6 $v\in W^{2m,q}(B)\subset
L^{\frac{Np}{N-(2m+1)p}}(B)$. Since $v=u$ on $B_{0}$ and $B_{0}$ was
arbitrary, we infer that $u\in
L^{\frac{Np}{N-(2m+1)p}}_{\mathrm{loc}}(\Omega)$.

If $p\geq\frac{N}{2m+1}$, we proceed in a similar fashion as in Case I. Fix
any $q\geq\frac{N}{N-2mp}$ and define $r:=\frac{Nq}{N+2mq}$. Then we get the
following relations

$1<r<\frac{N}{2m}\leq\frac{Np}{N-p}.$

We estimate as above using (2.7)

$\|f_{\varepsilon}+F(u,\eta)\|_{L^{r}(B)}\lesssim\|f_{\varepsilon}\|_{L^{\frac{N}{2m}}(B)}+\|u\|_{W^{2m,p}(B)}.$

We observe that the right hand side of (2.10) has finite $L^{r}(B)$ norm by
(2.9) and the hypothesis. Proposition 2.6 yields $v\in W^{2m,r}(B)\subset
L^{q}(B)$. Once again it implies that $u\in L^{q}(B_{0})$. Since
$B_{0}\subset\Omega$ and $q\geq\frac{N}{N-2mp}$ were arbitrary
$u\in\bigcap_{1\leq q<\infty}L^{q}_{\mathrm{loc}}(\Omega)$. ∎

We need the following observation which bears great resemblance to Proposition
2.6. We present the proof for the sake of the clarity.

######  Proposition 2.7.

Let $u\in W^{2m,p}_{\mathrm{loc}}(\Omega)$ satisfy $Lu\in
L^{q}_{\mathrm{loc}}(\Omega)$ for some $1<p<q<\infty$. Then $u\in
W^{2m,q}_{\mathrm{loc}}(\Omega)$.

###### Proof.

The proof is similar in spirit to that of Proposition 2.6. Fix
$B_{0}\subset\subset\Omega$ and choose another ball $B$ such that
$B_{0}\subset\subset B\subset\subset\Omega$. Let $\eta\in C^{\infty}_{0}(B)$
be smooth cut–off function such that $0\leq\eta\leq 1$ and $\eta=1$ on
$B_{0}$. We consider the function

$v:=\eta u\in W^{2m,p}(B)\cap W^{m,p}_{0}(B)$

which satisfies (as in (2.6))

$Lv=(Lu)\eta+\sum_{\beta<\alpha,\,|\alpha|=2m}c_{\alpha,\beta}(x)D^{\beta}uD^{\alpha-\beta}\eta=(Lu)\eta+F(u,\eta).$

We denote by $\tilde{A}_{s}$ the $L^{s}(B)$–realization of $L+\zeta I$.
Observe we only need to show that $v\in W^{2m,q}(B)$.

We consider two cases. If $W^{2m,p}(B)\subset W^{2m-1,q}(B)$ (that is either
$p>N$ or $q\leq\frac{Np}{N-p}$), then

$\|F(u,\eta)\|_{L^{q}(B)}\lesssim\|u\|_{W^{2m-1,q}(B)}\|\eta\|_{W^{2m,\infty}(B)}\lesssim\|u\|_{W^{2m,p}(B)},$

what yields

$Lv\in L^{q}(B).$

We choose $\zeta>0$ large enough so that both $\tilde{A}_{p}^{-1}$ and
$\tilde{A}_{q}^{-1}$ exist and set $f:=Lv+\zeta v\in L^{q}(B)$. We consider

$y=A_{q}^{-1}f\in D_{q}=W^{2m,q}\cap W^{m,q}_{0}(B)\subset D_{p}.$

Then we have $y-v\in D_{p}$ and

$\tilde{A}_{p}(y-v)=(L+\zeta)(y-v)=0$

Since $\tilde{A}_{p}:D_{p}\to L^{p}(B)$ is a bijection, we obtain $y=v$ and so
$v\in W^{2m,q}(B)$.

If $W^{2m,p}(B)\not\subset W^{2m-1,q}(B)$ (that is $p<N$ and
$q>\frac{Np}{N-p}$), then as in (2.6)

$\|F(u,\eta)\|_{L^{\frac{Np}{N-p}}(B)}\lesssim\|u\|_{W^{2m,p}(B)}.$

Hence

$Lv\in L^{\frac{Np}{N-p}}(B)$

and, similarly as above, this implies that $v\in D_{\frac{Np}{N-p}}$. After
finite number of such steps we find $s<N$ such that $v\in D_{s}\subset
W^{2m,s}(B)\subset L^{q}(B)$. Then the same argument as in the previous case
applies to show that $v\in W^{2m,q}(B)$.

Summarizing, in both cases $v\in W^{2m,q}(B)$ and the proof is completed. ∎

###### Proof of Theorem 2.1.

Let $p>1$ and $u\in W^{2m,p}_{\mathrm{loc}}(\Omega)$. If
$p\geq\frac{N}{2m+1}$, then $u\in\bigcap_{1\leq
q<\infty}L^{q}_{\mathrm{loc}}(\Omega)$ due to Lemma 2.5. We thus assume that
$p<\frac{N}{2m+1}$.

First, we show that whenever $u\in W^{2m,s}_{\text{loc}}(\Omega)$ with some
$1<s<\frac{N}{2m+1}$ is the strong solution of

$Lu=g(x,u)\quad\text{in }\Omega,$

then $u\in W^{2m,\frac{Ns}{N-s}}_{\text{loc}}(\Omega)$. Indeed, Lemma 2.5
implies that $u\in L^{\frac{Ns}{N-(2m+1)s}}_{\mathrm{loc}}(\Omega)$ and
$g(\cdot,u)\in L^{\frac{Ns}{N-s}}_{\mathrm{loc}}(\Omega)$ in view of
Proposition 2.3. Since $Lu=g(x,u)\in
L^{\frac{Ns}{N-s}}_{\mathrm{loc}}(\Omega)$, Proposition 2.7 yields $u\in
W^{2m,\frac{Ns}{N-s}}_{\mathrm{loc}}(\Omega)$ as claimed.

Let $k\geq 1$ be the integer satisfying

$k-1<\frac{N-(2m+1)p}{p}\leq k$

and define

$p_{i}:=\frac{Np}{N-ip},\quad\text{for }i=0,\ldots,k.$

Notice that $k$ is chosen in such a way that $p_{k-1}<\frac{N}{2m+1}$ and
$p_{k}\geq\frac{N}{2m+1}$. The first part of the proof shows that if $u\in
W^{2m,p_{i}}_{\mathrm{loc}}(\Omega)$, then $u\in
W^{2m,p_{i+1}}_{\mathrm{loc}}(\Omega)$. After $k$ such steps we obtain $u\in
W^{2m,p_{k}}_{\mathrm{loc}}(\Omega)$ and Lemma 2.5 yields the assertion.

We now proceed with the proof of the second assertion. Let $u\in
W^{2m,p}_{\mathrm{loc}}(\Omega)$ with $p>1$ be the strong solution of
$Lu=g(x,u)$ in $\Omega$ with $g$ of polynomial growth (2.3). If
$p\geq\frac{N}{2m}$, then the Sobolev embedding implies that
$u\in\bigcap_{1\leq 1<\infty}L^{q}_{\mathrm{loc}}(\Omega)$. Hence the
superposition $g(\cdot,u)\in L^{q}_{\mathrm{loc}}(\Omega)$ for any $q\geq 1$
and Proposition 2.7 yields $u\in W^{2m,q}_{\mathrm{loc}}(\Omega)$. Then $u\in
C^{2m-1,h}_{\mathrm{loc}}(\Omega)$ due to the Sobolev embeddings for any
$0<h<1$.

We now consider the case $p<\frac{N}{2m}$. We claim that there exists $a\in
L^{\frac{N}{2m}}_{\mathrm{loc}}(\Omega)$ such that

$|g(x,u(x))|\leq a(x)(1+|u(x)|),\quad\text{a.e. in }\Omega.$

Indeed, defining

(2.14) $a(x)=\frac{|g(x,u(x))|}{1+|u(x)|}$

and fixing some $B\subset\subset\Omega$ we have by (2.3)

$\displaystyle\int_{B}a^{\frac{N}{2m}}\,dx$
$\displaystyle\lesssim\int_{B}\left(\frac{1+|u|^{\frac{N}{N-2mp}}}{1+|u|}\right)^{\frac{N}{2m}}\,dx$
$\displaystyle\leq\int_{B\cap\left\\{x\mid|u(x)|\leq
1\right\\}}2^{\frac{N}{2m}}\,dx+\int_{B\cap\left\\{x\mid|u(x)|>1\right\\}}\left(\frac{2|u|^{\frac{N}{N-2mp}}}{|u|}\right)^{\frac{N}{2m}}\,dx$
$\displaystyle\lesssim
1+\int_{B}{|u|^{\frac{2mp}{N-2mp}\frac{N}{2m}}}\,dx<\infty,$

since $u\in W^{2m,p}(B)\subset L^{\frac{Np}{N-2mp}}(B)$. We apply the already
proved first part of Theorem 2.1 to infer that $u\in\bigcap_{1\leq
q<\infty}L^{q}_{\mathrm{loc}}(\Omega)$. This and the polynomial rate of growth
(2.3) enable us to conclude that $Lu=g(\cdot,u)\in
L^{q}_{\mathrm{loc}}(\Omega)$ for any $q\geq 1$. Once again Proposition 2.7
and the Sobolev embeddings finish the proof. ∎

##  3\. (Very) Weak Solutions

This section is devoted to the Brezis–Kato theorem for certain types of weak
solutions. We make stronger assumptions on the coefficients of the
differential operator $L$ to deal with less regular solutions. We consider the
differential operator in the divergence form as well.

Let us recall that $L$ is a strongly elliptic linear differential operator
satisfying the hypothesis stated in Introduction. We additionally assume here
that the coefficients $a_{\alpha}$ of $L$ are more regular, namely,

(3.1) $a_{\alpha}\in C^{|\alpha|}(\Omega),\quad\text{for}\;0<|\alpha|\leq 2m.$

Let $f\in L^{1}_{\text{loc}}(\Omega)$. We say that $u\in
L^{1}_{\text{loc}}(\Omega)$ is a _very weak solution_ of

$Lu=f,$

if for every $\varphi\in C^{\infty}_{0}(\Omega)$

$\int_{\Omega}u(x)\sum_{|\alpha|\leq
2m}(-1)^{m+|\alpha|}D^{\alpha}\left(a_{\alpha}(x)\varphi(x)\right)\,dx=\int_{\Omega}f(x)\varphi(x)\,dx.$

The above formula arises in a usual manner from formally multiplying the
differential equation $Lu=f$ by a test function $\varphi$, integrating over
$\Omega$ and using integration by parts.

######  Lemma 3.1.

Let $1<p,\,q<\infty$. If $u\in L^{q}_{\mathrm{loc}}(\Omega)$ is a very weak
solution of $Lu=f\in L^{p}_{\mathrm{loc}}(\Omega)$, then $u\in
W^{2m,p}_{\mathrm{loc}}(\Omega)$.

###### Proof.

Take any ball $B_{0}\subset\subset\Omega$ and choose another ball $B$ such
that $B_{0}\subset\subset B\subset\subset\Omega$. We introduce the adjoint $A$
of $L$ in $B$ by the formula

$Au:=\sum_{|\alpha|\leq
2m}(-1)^{m+|\alpha|}D^{\alpha}(\overline{a_{\alpha}(x)}u).$

Due to the Leibniz formula and (3.1) the operator $A$ is itself an elliptic
differential operator, i.e., $Au=\sum_{|\alpha|\leq
2m}b_{\alpha}(x)D^{\alpha}u$ for some coefficients $b_{\alpha}$. Note that $A$
and $u$ fulfill all the hypotheses of Thm 7.1 in [Agmon] and therefore $u\in
W^{2m,p}_{\mathrm{loc}}(B)\subset W^{2m,p}(B_{0})$. This finishes the proof. ∎

######  Theorem 3.2.

Let $L$ by strongly elliptic operator with the coefficients satisfying (3.1).
Let $u\in L^{p}_{\mathrm{loc}}(\Omega)$ for some $p>\frac{N}{N-2m}$ be a very
weak solution of

(3.2) $Lu=g(x,u)\quad\text{in }\Omega.$

If $g$ satisfies (2.2), then $u\in\bigcap_{1\leq
q<\infty}L^{q}_{\mathrm{loc}}(\Omega)$. If $g$ has a polynomial rate of
growth, namely,

(3.3) $|g(x,s)|\lesssim 1+|s|^{\frac{N+2mp}{N}},\quad\text{for all
}x\in\Omega\text{ and }s\in\mathbb{C},$

then $u\in W^{2m,q}_{\mathrm{loc}}(\Omega)\cap
C^{2m-1,h}_{\mathrm{loc}}(\Omega)$, for all $1\leq q<\infty$ and $0<h<1$.

###### Proof.

Let $u\in L^{p}_{\mathrm{loc}}(\Omega)$ be a very weak solution of
$Lu=g(x,u)$. By Proposition 2.3, $Lu=g(\cdot,u)\in
L^{\frac{Np}{N+2mp}}_{\mathrm{loc}}(\Omega)$ and $u\in
W^{2m,\frac{Np}{N+2mp}}_{\mathrm{loc}}(\Omega)$ in view of Lemma 3.1. Such a
regularity implies that $u$ is a strong solution of (3.2) and Theorem 2.1
yields $u\in\bigcap_{1\leq q<\infty}L^{q}_{\mathrm{loc}}(\Omega)$.

If $g$ satisfies (3.3), then we define

$a(x)=\frac{|g(x,u(x))|}{1+|u(x)|}$

and proceed similarly as in (2.14) and below to show that $a\in
L^{\frac{N}{2m}}_{\mathrm{loc}}(\Omega)$ and $g$ satisfies (2.2). By the first
part of the proof $u\in\bigcap_{1\leq q<\infty}L^{q}_{\mathrm{loc}}(\Omega)$
and the polynomial rate of growth (3.3) then implies that
$Lu=g(\cdot,u)\in\bigcap_{1\leq q<\infty}L^{q}_{\mathrm{loc}}(\Omega)$. We
deduce utilizing Lemma 3.1 that $u\in W^{2m,q}_{\mathrm{loc}}(\Omega)$ for
every $q\geq 1$. The last assertion follows by the Sobolev embeddings
$W^{2m,q}_{\mathrm{loc}}(\Omega)\subset C^{2m-1,h}_{\mathrm{loc}}(\Omega)$. ∎

We now consider a differential operator in a _divergence form_

(3.4) $Lu=(-1)^{m}\sum_{|\alpha|,\,|\beta|\leq
m}D^{\beta}\left(a_{\alpha\beta}(x)D^{\alpha}u\right),$

where $a_{\alpha\beta}\in C^{|\alpha|+|\beta|}(\Omega)$. We say that $L$ is
elliptic provided that

$\mathfrak{Re}\left(\sum_{|\alpha|=|\beta|=2m}a_{\alpha\beta}(x)\xi^{\alpha+\beta}\right)\geq\lambda|\xi|^{2m},\quad\text{for
}\xi\in\mathbb{R}^{N}.$

Let $f\in L^{1}_{\mathrm{loc}}(\Omega)$. We say that $u\in
W^{m,1}_{\mathrm{loc}}(\Omega)$ is a _weak solution_ of

$Lu=f\quad\text{in }\Omega,$

if for every test function $\varphi\in C^{\infty}_{0}(\Omega)$

$\int_{\Omega}\sum_{|\alpha|,\,|\beta|\leq
m}(-1)^{m+|\beta|}a_{\alpha\beta}(x)D^{\alpha}u(x)D^{\beta}\varphi(x)\,dx=\int_{\Omega}f(x)\varphi(x)\,dx.$

######  Theorem 3.3.

Let $L$ be the elliptic differential operator in the divergence form (3.4)
with the coefficients $a_{\alpha\beta}\in C^{|\alpha|+|\beta|}(\Omega)$. Let
$u\in W^{m,p}_{\mathrm{loc}}(\Omega)$ with $p>\frac{N}{N-m}$ be a weak
solution of

$Lu=g(x,u)\quad\text{in }\Omega.$

If $g$ satisfies (2.2), then $u\in\bigcap_{1\leq
q<\infty}L^{q}_{\mathrm{loc}}(\Omega)$. If $g$ has the polynomial rate of
growth: there is $r>0$ such that

(3.5) $|g(x,s)|\lesssim 1+|s|^{r},\quad\text{for all }x\in\Omega\text{ and
}s\in\mathbb{C},\text{ where }r=\begin{cases}\frac{N+mp}{N-mp},&\text{if
}p<\frac{N}{m},\\\ \text{is arbitrary},&\text{if
}p\geq\frac{N}{m},\end{cases}$

then $u\in W^{2m,q}_{\mathrm{loc}}(\Omega)\cap
C^{2m-1,h}_{\mathrm{loc}}(\Omega)$ for all $1\leq q<\infty$ and $0<h<1$.

###### Proof.

If $p\geq\frac{N}{m}$, then $u\in\bigcap_{1\leq
q<\infty}L^{q}_{\mathrm{loc}}(\Omega)$ due to the Sobolev embedding. Hence we
assume that $p<\frac{N}{m}$. Fix $\varphi\in C^{\infty}_{0}(\Omega)$ in the
definition of a weak solution. Using integration by parts and ordering the
terms one may show that $u$ is also a very weak solution to some differential
operator in the non-divergence form $\tilde{L}u=(-1)^{m}\sum_{|\alpha|\leq
2m}c_{\alpha}(x)D^{\alpha}u=g(x,u)$ in $\Omega$. The smoothness assumption on
the coefficients $a_{\alpha\beta}\in C^{|\alpha|+|\beta|}(\Omega)$ implies
that $c_{\alpha}\in C^{|\alpha|}(\Omega)$. Moreover, we have
$\sum_{|\alpha|=2m}c_{\alpha}(x)\xi^{\alpha}=\sum_{|\alpha|=|\beta|=m}a_{\alpha\beta}(x)\xi^{\alpha+\beta}$,
so $\tilde{L}$ is elliptic as well. The Sobolev embedding
$W^{m,p}_{\mathrm{loc}}(\Omega)\subset
L^{\frac{Np}{N-mp}}_{\mathrm{loc}}(\Omega)$ where
$\frac{Np}{N-mp}>\frac{N}{N-2m}$ and Theorem 3.2 proves the assertion.

Let us now assume that (3.5) holds. If $p\geq\frac{N}{m}$, then the Sobolev
embedding yields $u\in\bigcap_{1\leq q<\infty}L^{q}_{\mathrm{loc}}(\Omega)$.
If $p<\frac{N}{m}$, then in view of (3.5) $g$ satisfies (2.2) with
$a(x):=\frac{|g(x,u)|}{1+|u|}$ what may be shown as above. Therefore, in any
case we obtain $u\in\bigcap_{1\leq q<\infty}L^{q}_{\mathrm{loc}}(\Omega)$. The
polynomial rate of growth implies that $g(\cdot,u)\in\bigcap_{1\leq
q<\infty}L^{q}_{\mathrm{loc}}(\Omega)$, so $\tilde{L}u\in\bigcap_{1\leq
q<\infty}L^{q}_{\mathrm{loc}}(\Omega)$. Lemma 3.1 and the Sobolev embeddings
finishes the proof. ∎

We present a particular form of Theorem 3.3 for polyharmonic operators in a
(most common) $L^{2}$ setting for future references.

######  Corollary 3.4.

Let $L$ be the polyharmonic operator $(-\Delta)^{m}$ defined on an open set
$\Omega\subset\mathbb{R}^{N}$ with $N>2m$. Let $u\in
W^{m,2}_{\mathrm{loc}}(\Omega)$ be a weak solution of

$(-\Delta)^{m}u=g(x,u)\quad\text{in }\Omega,$

in the sense that for every $\varphi\in C^{\infty}_{0}(\Omega)$

$\displaystyle\int_{\Omega}\Delta^{k}u\Delta^{k}\varphi\,dx=\int_{\Omega}g(x,u)\varphi\,dx,$
$\displaystyle\text{if}\ m=2k,$
$\displaystyle\int_{\Omega}\nabla\Delta^{k}u\nabla\Delta^{k}\varphi\,dx=\int_{\Omega}g(x,u)\varphi\,dx,$
$\displaystyle\text{if}\ m=2k+1.$

If $g$ satisfies (2.2), then $u\in\bigcap_{1\leq
q<\infty}L^{q}_{\mathrm{loc}}(\Omega)$. If $g$ has a polynomial rate of
growth:

$|g(x,s)|\lesssim 1+|s|^{\frac{N+2m}{N-2m}},\quad\text{for all
}x\in\Omega\text{ and }s\in\mathbb{C},$

then $u\in W^{2m,q}_{\mathrm{loc}}(\Omega)\cap
C^{2m-1,h}_{\mathrm{loc}}(\Omega)$ for all $1\leq q<\infty$ and $0<h<1$.

## Acknowledgements

The research was partially supported by Narodowe Centrum Nauki Grant No.
2017/26/E/ST1/00817.

## References