An Existence Criterion for Maximizers of Convolution Operators in $$\boldsymbol{L}_{\mathbf{1}}\boldsymbol{(\mathbb{R}^{n})}$$

Kalachev, Gleb; Sadov, Sergey

doi:10.3103/s0027132221040033

Cited by 2 publications

(2 citation statements)

References 9 publications

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“…The "boundary" cases p = 1 , r = ∞ , and p = r are discussed in [1], too. The full analysis of the case p = q = r = 1 is given in [9]. The present paper is motivated by a desire to connect Theorem 1 with another maximizer existence result due to Lieb [10], see also [7,Sec.…”

Section: Introductionmentioning

confidence: 99%

EXISTENCE OF CONVOLUTION MAXIMIZERS IN $$L_p(\mathbb {R}^n)$$ WITH KERNELS FROM LORENTZ SPACES

Sadov¹

2023

J Math Sci

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The paper extends an earlier result of G.V. Kalachev et al. (Sb. Math. 210(8):1129-1147, 2019 on the existence of a maximizer of convolution operator acting between two Lebesgue spaces on ℝ n with kernel from some L q , 1 < q < ∞ . On the other hand, E. Lieb (Ann. of Math. 118:(2): 1983) proved the existence of a maximizer for the Hardy-Littlewood-Sobolev inequality and remarked that in general a convolution maximizer for a kernel from weak L q may not exist. In this paper we axiomatize some properties used in the proof of the Kalachev-Sadov 2019 theorem and obtain a more general result. As a consequence, we prove that the convolution maximizer always exists for kernels from a slightly more narrow class than weak L q , which contains all Lorentz spaces L q,s with q ≤ s < ∞.

show abstract

Section: Introductionmentioning

confidence: 99%

EXISTENCE OF CONVOLUTION MAXIMIZERS IN $$L_p(\mathbb {R}^n)$$ WITH KERNELS FROM LORENTZ SPACES

Sadov¹

2023

J Math Sci

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show abstract

“…The "boundary" cases p = 1, r = ∞, and p = r are discussed in [2], too. The full analysis of the case p = q = r = 1 is given in [3].…”

Section: Introductionmentioning

confidence: 99%

Existence of convolution maximizers in $L_p(R^n)$ for kernels from Lorentz spaces

Sadov¹

2022

Preprint

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The paper extends an earlier result of G.V. Kalachev and the author (Sb. Math. 2019) on the existence of a maximizer of convolution operator acting between two Lebesgue spaces on R n with kernel from some L q , 1 < q < ∞. In view of Lieb's result of 1983 about the existence of an extremizer for the Hardy-Littlewood-Sobolev inequality it is natural to ask whether a convolution maximizer exists for any kernel from weak L q . The answer in the negative was given by Lieb in the above citation. In this paper we prove the existence of maximizers for kernels from a slightly more narrow class than weak L q , which contains all Lorentz spaces L q,s with q ≤ s < ∞.

show abstract

An Existence Criterion for Maximizers of Convolution Operators in $$\boldsymbol{L}_{\mathbf{1}}\boldsymbol{(\mathbb{R}^{n})}$$

Cited by 2 publications

References 9 publications

EXISTENCE OF CONVOLUTION MAXIMIZERS IN $$L_p(\mathbb {R}^n)$$ WITH KERNELS FROM LORENTZ SPACES

EXISTENCE OF CONVOLUTION MAXIMIZERS IN $$L_p(\mathbb {R}^n)$$ WITH KERNELS FROM LORENTZ SPACES

Existence of convolution maximizers in $L_p(R^n)$ for kernels from Lorentz spaces

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