On maximizers of a convolution operator in -spaces

Abstract

We consider a convolution operator in R d with kernel in L q acting from L p to L s , where 1/p + 1/q = 1 + 1/s. The main theorem states that if 1 < q, p, s < ∞, then there exists an L p function of unit norm on which the s-norm of the convolution is attained. A number of questions, related to the statement and proof of the main theorem, are discussed. Also the problem of computing best constants in the Hausdorff-Young inequality for the Laplace transform, which prompted this research, is considered.1 Emphasizing that K p,r is the extremum of the symmetric bilinear form K p,r = K r,p = sup k(x + y)f (x)g(y) dx dy ; f p = g r = 1. 2 The labels (a) , (b) etc. refer to the comments (a), (b) etc. in Section 8. 1. The class Max comprises all maximizing sequences (for the convolution operator K : L p → L r ). 2. The class SMax comprises all special maximizing sequences, that is, maximizing sequences of the form (Bf n ), where (f n ) ∈ Max. 3. The class RTgt comprises all relatively tight sequences. 4. The class Tgt comprises all tight sequences. 5. The class WCvg comprises all weakly convergent sequences. 6. The class LCvg comprises all locally convergent sequences, i.e. sequences converging in L p norm on any bounded measurable subset of R d .

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

“…The main result of [2], sharpening the earlier result of Pearson [8] (by removing extraneous assumptions), is Theorem A. If p, q, r are related by (1), 1 < p < r < ∞, and k ∈ L q , then the operator K k : L p → L r has a maximizer.…”

“…In this paper we extend the main result of [2]. We will mostly follow [2] in notation and terminology.…”

“…Convolution operators considered in this paper will have kernels from Cnv(p, r) where 1 < p < r < ∞. (In [2], the exponent of the target space was denoted r ′ . )…”

“…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].…”

“…In comparison with [2], the novelty in this paper is primarily in formulations. For the proof of the base result, Theorem 1, we have to make only modest adjustments of some lemmas from [2].…”