A note on the trace inequality for Riesz potentials

Giorgi Imerlishvili; Alexander Meskhi

doi:10.1515/gmj-2020-2077

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A note on the trace inequality for Riesz potentials

Giorgi Imerlishvili und Alexander Meskhi

Veröffentlicht/Copyright: 7. November 2020

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Aus der Zeitschrift Georgian Mathematical Journal Band 28 Heft 5

Abstract

We establish a necessary and sufficient condition on a non-negative locally integrable function v guaranteeing the (trace) inequality

∥ I α ⁢ f ∥ L v p ⁢ ( ℝ n ) ≤ C ⁢ ∥ f ∥ L p , 1 ⁢ ( ℝ n )

for the Riesz potential I α , where L p , 1 ⁢ ( ℝ n ) is the Lorentz space. The same problem is studied for potentials defined on spaces of homogeneous type.

Keywords: Riesz potential; trace inequality; weight

MSC 2010: 26A33; 42B35

1 Introduction

Trace inequalities for Riesz potentials I α deal with non-negative measures ν such that

(1.1) ( ∫ ℝ n | I α ⁢ f ⁢ ( x ) | q ⁢ 𝑑 ν ⁢ ( x ) ) 1 / q ≤ C ⁢ ( ∫ ℝ n | f ⁢ ( x ) | p ⁢ 𝑑 x ) 1 / p .

Adams [1] proved that the necessary and sufficient condition on ν guaranteeing (1.1) for 1 < p < q < ∞ and 0 < α < n p is that the measure ν satisfies the following condition: there is a positive constant C such that for all balls B ⊂ ℝ n ,

ν ⁢ ( B ) ≤ C ⁢ | B | ( α / n - 1 / p ) ⁢ q .

The Riesz potential operator

I α ⁢ f ⁢ ( x ) = ∫ ℝ n f ⁢ ( y ) | x - y | n - α ⁢ 𝑑 y , 0 < α < n , x ∈ ℝ n ,

plays an important role in the investigation of PDEs. It is worth mentioning its role in the theory of Sobolev’s embeddings (see, e.g., [14]).

An appropriate fractional maximal operator is given by the formula

M α ⁢ f ⁢ ( x ) = sup B ∋ x ⁡ 1 | B | 1 - α / n ⁢ ∫ B | f ⁢ ( y ) | ⁢ 𝑑 y , 0 ≤ α < n , x ∈ ℝ n .

Here, M 0 ⁢ f = M ⁢ f is the Hardy–Littlewood maximal function having much importance in Harmonic Analysis, in particular, in the theory of singular integrals (see, e.g., [18]).

Let v be a non-negative locally integrable function on ℝ n . We are interested in inequality (1.1) for d ⁢ ν ⁢ ( x ) = v ⁢ ( x ) ⁢ d ⁢ x , i.e.,

(1.2) ∥ I α ⁢ f ∥ L v q ⁢ ( ℝ n ) ≤ C ⁢ ∥ f ∥ L p ⁢ ( ℝ n ) .

In this case by the result of Adams [1], the condition

(1.3) [ v ] p , q , α := sup B ⁡ ( v ⁢ ( B ) ) 1 / q ⁢ | B | α / n - 1 / p < ∞ ,

where the supremum is taken over all balls B ⊂ ℝ n , is simultaneously necessary and sufficient whenever 1 < p < q < ∞ and 0 < α < n p . In the case p = q the implication (1.2) ⟹ (1.3) can be checked easily by considering the test functions χ B . However, (1.3) ⟹ (1.2) is not true (see the counterexamples in [2, 8] for a measure ν, and [10] for a non-negative function v).

Our aim is to find a Lorentz space L p , s which is narrower than the class L p ⁢ ( ℝ n ) (i.e., s < p ) and for which the inequality

(1.4) ∥ I α ⁢ f ∥ L v p ⁢ ( ℝ n ) ≤ C ⁢ ∥ f ∥ L p , s ⁢ ( ℝ n )

holds if and only if (1.3) is satisfied for p = q . In particular, we show that (1.4) is equivalent to condition (1.3) for s = 1 . The question whether this holds for 1 < s < p remains open.

As is known, various criteria were obtained for (1.2) with p = q (see [2, 12, 13, 8, 15]). For the solution of the two-weight problem for Riesz potential operators I α , we refer to [5, 17] (see also the monograph [9]).

Inequality (1.2) for p = q implies the estimate

∥ f ∥ L v q ⁢ ( ℝ n ) ≤ C ⁢ ∥ ∇ ⁡ f ∥ L p ⁢ ( ℝ n ) , f ∈ C 0 ∞ ,

which follows from the estimate

| f ⁢ ( x ) | ≤ C ⁢ I 1 ⁢ ( | ∇ ⁡ f | ) ⁢ ( x ) .

The following Fefferman–Phong-type theorem [4] holds true.

Theorem A.

Let 1 < p < ∞ and 0 < α < n p . Then the inequality

∥ I α ⁢ f ∥ L v p ≤ C ⁢ [ v ] p , r , α * ⁢ ∥ f ∥ L p

is fulfilled for some p < r , where

(1.5) [ v ] p , r , α * := sup B ⁡ | B | α / n - 1 / r ⁢ ( ∫ B v r / p ⁢ ( x ) ⁢ 𝑑 x ) 1 / r < ∞ .

Remark 1.1.

It is easy to see that by Hölder’s inequality we have that condition (1.5) is stronger than (1.3) for p = q , in particular, [ v ] p , α ≤ [ v ] p , r , α * for r > p , where [ v ] p , α = [ v ] p , p , α .

2 Preliminaries

Let f be a measurable function on ℝ n and let 1 ≤ p < ∞ , 1 ≤ s ≤ ∞ .

We say that f belongs to the Lorentz space L p , s if

∥ f ∥ L p , s = { ( s ⁢ ∫ 0 ∞ ( | { x ∈ ℝ n : | f ⁢ ( x ) | > τ } | ) s / p ⁢ τ s - 1 ⁢ 𝑑 τ ) 1 / s if ⁢ 1 ≤ s < ∞ , sup s > 0 ⁡ s ⁢ ( | { x ∈ ℝ n : | f ⁢ ( x ) | > s } | ) 1 / p if ⁢ s = ∞

is finite.

If p = s , then L p , s coincides with the Lebesgue space L p .

Denote by f * a non-increasing rearrangement of f. Then, by integration by parts, it can be checked that (see also [7])

∥ f ∥ L p , s = { ( s p ⁢ ∫ 0 ∞ ( t 1 / p ⁢ f * ⁢ ( t ) ) s ⁢ d ⁢ t t ) 1 / s if ⁢ 1 ≤ s < ∞ , sup t > 0 ⁡ { t 1 / p ⁢ f * ⁢ ( t ) } if ⁢ s = ∞ .

Now we list some useful properties of Lorentz spaces (see, e.g., [7]):

∥ χ E ∥ L p , s = | E | 1 / p .
If 1 ≤ p < ∞ and s 2 ≤ s 1 , then L p , s 2 ↪ L p , s 1 with the embedding constant C p , s 1 , s 2 depending only on p, s 1 and s 2 .
There is a positive constant C p , s such that

C p , s - 1 ⁢ ∥ f ∥ L p , s ≤ sup ∥ h ∥ L p ′ , s ′ ≤ 1 ⁡ | ∫ X f ⁢ ( x ) ⁢ h ⁢ ( x ) ⁢ 𝑑 μ ⁢ ( x ) | ≤ C p , s ⁢ ∥ f ∥ L p , s

for every f ∈ L w p , s , where p ′ = p / ( p - 1 ) , s ′ = s / ( s - 1 ) .
(Hölder’s inequality) Let 1 p = 1 p 1 + 1 p 2 and 1 s = 1 s 1 + 1 s 2 . Then

∥ f 1 ⁢ f 2 ∥ L p , s ≤ C ⁢ ∥ f 1 ∥ L p 1 , s 1 ⁢ ∥ f 2 ∥ L p 2 , s 2

for all f 1 ∈ L p 1 , s 1 and f 2 ∈ L p 2 , s 2 , where C = C p , s , p 1 , p 2 , s 1 , s 2 .

In the sequel, for a non-negative function v and a set E the symbol v ⁢ ( E ) will denote the integral

v ⁢ ( E ) := ∫ E v ⁢ ( x ) ⁢ 𝑑 x .

By the symbol A ≈ B we mean that there exists a positive constant c such that

c - 1 ⁢ A ≤ B ≤ c ⁢ A .

3 Main result

The main result of this note reads as follows.

Theorem 3.1.

Let 1 < p < ∞ and 0 < α < n p . Suppose that v is a non-negative locally integrable function on R n . Then the following statements are equivalent:

There is a positive constant C such that for all f ∈ L p , 1 ⁢ ( ℝ n ) ,

(3.1) ∥ I α ⁢ f ∥ L v p ⁢ ( ℝ n ) ≤ C ⁢ ∥ f ∥ L p , 1 ⁢ ( ℝ n ) .
There is a positive constant c such that for all f ∈ L p , 1 ⁢ ( ℝ n ) ,

(3.2) ∥ M α ⁢ f ∥ L v p ⁢ ( ℝ n ) ≤ c ⁢ ∥ f ∥ L p , 1 ⁢ ( ℝ n ) .
[ v ] p , α = sup B ⁡ ( v ⁢ ( B ) ) 1 / p ⁢ | B | α / n - 1 / p < ∞ .

Moreover, if C and c are best constants in ( 3.1 ) and ( 3.2 ), respectively, then

C ≈ c ≈ [ v ] p , α .

To prove this theorem we need some lemmas.

For a non-negative a.e. finite function w we set E w := { x ∈ ℝ : w ⁢ ( x ) ≠ 0 } .

We also set p ′ = p p - 1 for 1 ≤ p ≤ ∞ .

Lemma 3.1.

Let 1 < p < ∞ and 0 < α < n / p ′ . Then the condition

[ w ] p , α ′ ≡ sup B ⁡ ( ∫ B w 1 - p ′ ⁢ ( x ) ⁢ χ E w ⁢ ( x ) ⁢ 𝑑 x ) 1 p ′ ⁢ | B | α n - 1 p ′ < ∞

guarantees that there is a positive constant C = C α , p such that for all f ∈ L w p ⁢ ( E w ) ,

∥ M α ⁢ ( f ⁢ χ E w ) ∥ L p , ∞ ⁢ ( ℝ n ) ≤ C ⁢ ∥ f ⁢ χ E w ∥ L w p ⁢ ( ℝ n ) .

Proof.

Observe that By Hölder’s inequality we find that

1 | B | 1 - α n ⁢ ∫ B | f ⁢ ( y ) | ⁢ χ E w ⁢ ( y ) ⁢ 𝑑 y = 1 | B | 1 - α n ⁢ ∫ B | f ⁢ ( y ) | ⁢ χ E w ⁢ ( y ) ⁢ w 1 p ⁢ ( y ) ⁢ w - 1 p ⁢ ( y ) ⁢ 𝑑 y

≤ 1 | B | 1 - α n ⁢ ( ∫ B | f ⁢ ( y ) | p ⁢ w ⁢ ( y ) ⁢ χ E w ⁢ ( y ) ⁢ 𝑑 y ) 1 p ⁢ ( ∫ B w 1 - p ′ ⁢ ( y ) ⁢ χ E w ⁢ ( y ) ⁢ 𝑑 y ) 1 p ′

≤ [ w ] p , α ′ ⁢ ( 1 | B | ⁢ ∫ B | f ⁢ ( y ) | p ⁢ χ E w ⁢ w ⁢ ( y ) ⁢ 𝑑 y ) 1 p

= [ w ] p , α ′ ⁢ ( M ⁢ ( | f | p ⁢ χ E w ⁢ w ) ) 1 p ,

where M is the Hardy–Littlewood maximal operator.

Hence

M α ⁢ ( f ⁢ χ E w ) ⁢ ( x ) ≤ [ w ] p , α ′ ⁢ ( M ⁢ ( | f | p ⁢ χ E w ⁢ w ) ) 1 p .

Consequently,

| { x ∈ ℝ n : M α ⁢ ( f ⁢ χ E w ) ⁢ ( x ) > λ } | ≤ | { x ∈ ℝ n : C ⁢ [ w ] p , α ′ ⁢ ( M ⁢ ( | f ⁢ χ E w | p ⁢ w ) ⁢ ( x ) ) 1 p > λ } |

≤ C ⁢ [ w ] p , α ′ λ p ⁢ ∥ | f | p ⁢ χ E w ⁢ w ∥ L 1

= C ⁢ [ w ] p , α ′ λ p ⁢ ∥ f ∥ L w p ⁢ ( E w ) p . ∎

Lemma 3.2.

Let 1 < p < ∞ and 0 < α < n / p ′ . If [ w ] p , α ′ < ∞ (see Lemma 3.1), then there is a positive constant C = C p , α such that for all f ∈ L w p ⁢ ( E w ) ,

∥ I α ⁢ ( f ⁢ χ E w ) ∥ L p , ∞ ⁢ ( ℝ n ) ≤ C p , α ⁢ ∥ f ⁢ χ E w ∥ L w p ⁢ ( ℝ n ) .

Proof.

It is known (see, e.g., [16, Theorem 1] for a more general setting) that

∥ I α ⁢ g ∥ L p , ∞ ⁢ ( ℝ n ) ≈ ∥ M α ⁢ g ∥ L p , ∞ ⁢ ( ℝ n )

for a measurable function g. Thus, by Lemma 3.1, the result follows. ∎

Lemma 3.3.

Let 1 < p < ∞ and 0 < α < n / p ′ . If [ w ] p , α ′ < ∞ , then there is a positive constant C = C p , α such that for all f ∈ L p ′ , 1 ⁢ ( E w ) ,

∥ I α ⁢ f ∥ L w 1 - p ′ p ′ ⁢ ( E w ) ≤ C ⁢ [ w ] p , α ′ ⁢ ∥ f ∥ L p ′ , 1 ⁢ ( E w ) .

Proof.

For f ≥ 0 , we have

∥ I α ⁢ f ∥ L w 1 - p ′ p ′ ⁢ ( E w ) = sup ∥ g ∥ L w p ⁢ ( E w ) ≤ 1 ⁡ | ∫ ℝ n ( I α ⁢ f ) ⁢ ( x ) ⁢ g ⁢ ( x ) ⁢ w - 1 p ⁢ ( x ) ⁢ w 1 p ⁢ ( x ) ⁢ χ E w ⁢ ( x ) ⁢ 𝑑 x |

≤ sup ∥ g ∥ L w p ⁢ ( E w ) ≤ 1 ⁡ ( ∫ ℝ n ( I α ⁢ f ) ⁢ ( x ) ⁢ | g ⁢ ( x ) | ⁢ χ E w ⁢ ( x ) ⁢ 𝑑 x )

= sup ∥ g ∥ L w p ⁢ ( E w ) ≤ 1 ⁡ ∫ E w I α ⁢ ( | g | ⁢ χ E w ) ⁢ ( y ) ⁢ f ⁢ ( y ) ⁢ 𝑑 y

≤ sup ∥ g ∥ L w p ⁢ ( E w ) ≤ 1 ⁡ ∥ I α ⁢ | g | ∥ L p , ∞ ⁢ ( E w ) ⁢ ∥ f ∥ L p ′ , 1 ⁢ ( E w )

≤ C p , α ⁢ [ w ] p , α ′ ⁢ ∥ f ∥ L p ′ , 1 ⁢ ( E w ) . ∎

Proof of Theorem 3.1.

(iii) ⟹ (i): Observe that if 1 < p < ∞ , 0 < α < n p , and v is a non-negative function satisfying [ v ] p , α < ∞ , then | { x ∈ ℝ n : v ⁢ ( x ) = ∞ } | = 0 . Hence

| { x ∈ ℝ n : v 1 - p ⁢ ( x ) = 0 } | = 0 .

Consequently, since

[ v ] p , α = sup B ⁡ ( ∫ B v ( 1 - p ) ⁢ ( 1 - p ′ ) ⁢ ( x ) ⁢ 𝑑 x ) 1 p ⁢ | B | α n - 1 p < ∞ ,

from Lemma 3.3 we have

∥ I α ⁢ f ∥ L v p ⁢ ( ℝ n ) ≤ C p , α ⁢ [ v ] p , α ⁢ ∥ f ∥ L p , 1 ⁢ ( ℝ n ) .

The pointwise estimate

M α ⁢ g ⁢ ( x ) ≤ I α ⁢ g ⁢ ( x ) , g ≥ 0 ,

implies the implication (i) ⟹ (ii).

Taking the test functions

f B 0 = χ B 0

in (3.2), where B 0 is a ball in ℝ n , using the estimate

M α ⁢ f B 0 ⁢ ( x ) ≥ C n , α ⁢ | B 0 | α n , x ∈ B 0 ,

and observing that

∥ f B 0 ∥ L p , 1 = c ⁢ | B 0 | 1 p ,

we can easily see that (ii) implies

v ⁢ ( B 0 ) 1 / p ≤ C ⁢ | B 0 | 1 / p - α / n ,

with a positive constant C independent of B 0 . The theorem has been proved. ∎

4 The case of spaces of homogeneous type

Let ( X , d , μ ) be a quasi-metric measure space with a quasi-metric d and measure μ. A quasi-metric d is a function d : X × X → [ 0 , ∞ ) which satisfies the following conditions:

d ⁢ ( x , y ) = 0 if and only if x = y .
For all x , y ∈ X , one has d ⁢ ( x , y ) = d ⁢ ( y , x ) .
There is a positive constant κ such that d ⁢ ( x , y ) ≤ κ ⁢ ( d ⁢ ( x , z ) + d ⁢ ( z , y ) ) for all x , y , z ∈ X .

In the sequel, we will assume that the balls B ⁢ ( x , r ) := { y ∈ X : d ⁢ ( x , y ) < r } are measurable with positive measure μ for all x ∈ X and r > 0 .

If μ satisfies the doubling condition

μ ⁢ ( B ⁢ ( x , 2 ⁢ r ) ) ≤ C μ ⁢ μ ⁢ ( B ⁢ ( x , r ) ) ,

with a positive constant C μ independent of x and r, then we say that ( X , d , μ ) is a space of homogeneous type (SHT). Throughout the paper, we assume that ( X , d , μ ) is an SHT.

For example, rectifiable curves in ℂ with Euclidean distance and arc-length measure satisfying the Carleson (regularity) condition, nilpotent Lie groups with Haar measure, and domains in ℝ n with the so-called 𝒜 condition are examples of an SHT. For the definition, examples and some properties of an SHT, see, e.g., the paper [11] and the monographs [19, 3].

For a given quasi-metric measure space ( X , d , μ ) and q satisfying 1 ≤ q ≤ ∞ , as usual, we will denote by L q = L q ⁢ ( X , μ ) the Lebesgue space equipped with the standard norm. Let L p , s ⁢ ( X , μ ) be the Lorentz space defined on an SHT ( X , d , μ ) .

Let us denote by K α ⁢ f the Riesz potential of a μ-measurable function f defined by the formula

K α ⁢ f ⁢ ( x ) = ∫ X μ ⁢ ( B x ⁢ y ) α - 1 ⁢ f ⁢ ( y ) ⁢ 𝑑 μ ⁢ ( y ) , x ∈ X ,

where 0 < α < 1 and B x ⁢ y := B ⁢ ( x , d ⁢ ( x , y ) ) .

An appropriate fractional maximal function has the form

ℳ α ⁢ f ⁢ ( x ) = sup B ∋ x ⁡ 1 μ ⁢ ( B ) 1 - α ⁢ ∫ B | f ⁢ ( y ) | ⁢ 𝑑 μ ⁢ ( y ) , x ∈ X .

The following trace inequality for an SHT was proved by Gabidzashvili (see [6]).

Theorem B.

Let 1 < p < q < ∞ and 0 < α < 1 p . Suppose that ( X , d , μ ) is an SHT and ν is another measure on X.Then the inequality

∥ K α ⁢ f ∥ L q ⁢ ( X , ν ) ≤ C ⁢ ∥ f ∥ L p ⁢ ( X , μ )

holds if and only if

sup B ⁡ ν ⁢ ( B ) 1 / q ⁢ μ ⁢ ( B ) α - 1 / p < ∞ ,

where the supremum is taken over all balls B ⊂ X .

Using the proof of Theorem 3.1, we can formulate the same result for an SHT. In particular, the following theorem holds true.

Theorem 4.1.

Let 1 < p < ∞ and 0 < α < 1 p . Suppose that ( X , d , μ ) is an SHT. Assume that v is a non-negative μ-locally integrable a.e. finite function on X. Then the following statements are equivalent:

There is a positive constant C such that for all f ∈ L p , 1 ⁢ ( X , μ ) ,

(4.1) ∥ K α ⁢ f ∥ L v p ⁢ ( X , μ ) ≤ C ⁢ ∥ f ∥ L p , 1 ⁢ ( X , μ ) .
There is a positive constant c such that for all f ∈ L p , 1 ⁢ ( X , μ ) ,

(4.2) ∥ ℳ α ⁢ f ∥ L v p ⁢ ( X , μ ) ≤ c ⁢ ∥ f ∥ L p , 1 ⁢ ( X , μ ) .
[ v ] p , α , X , μ = sup B ⁡ ( ∫ B v ⁢ ( x ) ⁢ 𝑑 μ ⁢ ( x ) ) 1 / p ⁢ μ ⁢ ( B ) α - 1 / p < ∞ .

Moreover, if C and c are best constants in ( 4.1 ) and ( 4.2 ), respectively, then

C ≈ c ≈ [ v ] p , α , X , μ .

Funding source: Shota Rustaveli National Science Foundation

Award Identifier / Grant number: FR-18-2499

Funding statement: This work was supported by the Shota Rustaveli National Science Foundation of Georgia (Project no. FR-18-2499).

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Received: 2019-09-18

Accepted: 2020-01-09

Published Online: 2020-11-07

Published in Print: 2021-10-01

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