Unions et intersections d'espaces $L^p$ invariantes par translation ou convolution

Bertrandias, Jean-Paul; Datry, Christian; Dupuis, Christian

doi:10.5802/aif.689

Cited by 39 publications

(21 citation statements)

References 11 publications

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“…That can also be shown directly, using Theorem 2 (for similar assertions cf. [1], [2], [7] or [12]). Corollary 4.…”

Section: ])mentioning

confidence: 99%

A characterization of minimal homogeneous Banach spaces

Feichtinger¹

1981

Proc. Amer. Math. Soc.

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Abstract. Let G be a locally compact group. It is shown that for a homogeneous Banach space B on G satisfying a slight additional condition there exists a minimal space fimm in the family of all homogeneous Banach spaces which contain all elements of B with compact support. Two characterizations of Bm¡1¡ aie given, the first one in terms of "atomic" representations. The equivalence of these two characterizations is derived by means of certain (bounded) partitions of unity which are of interest for themselves.

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“…That can also be shown directly, using Theorem 2 (for similar assertions cf. [1], [2], [7] or [12]). Corollary 4.…”

Section: ])mentioning

confidence: 99%

A characterization of minimal homogeneous Banach spaces

Feichtinger¹

1981

Proc. Amer. Math. Soc.

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“…Kellogg [10] and Williams [17] studied analogous spaces on certain discrete groups. The paper [18] of Bertrandias, Datry, and Depuis appeared while this paper was in a referee's hands. It independently develops, for abelian groups G, spaces lq(Lp) equivalent to our mixed-norm spaces L(pq)(G).…”

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confidence: 99%

“…Our techniques for nonabelian G require the uniform partition and mixed-norm structure as do some spaces we use which are not equivalent to those of [18] even for abelian G. We have therefore not adopted the elegant approach of [18], which was developed for different ends.…”

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confidence: 99%

“…functions are the closures in L(poo)(R) of the trigonometric polynomials.) Particular mixed-norm spaces on R" studied by N. Wiener, R. Goldberg, and P. Szeptycki are cited in [18]. §4 investigates convolution of functions from mixed-norm spaces and generalizes Young's inequality.…”

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confidence: 99%

“…Theorem 4.2 (cf. [18]). Let f G L/, g G L", T < p_, q < öö, l/p2 + l/q2 > 1; then (a)/ * g exists (and is finite) locally a.e.…”

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confidence: 99%

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Product-Convolution Operators and Mixed-Norm Spaces

Busby¹,

Smith²

1981

Transactions of the American Mathematical Society

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Abstract.Conditions for boundedness and compactness of product-convolution operators g -» PhCß = h ■ (/» g) on spaces L^G) are studied. It is necessary for boundedness to define a class of "mixed-norm" spaces L,p>q){G) interpolating the Lp(G) spaces in a natural way (L^^ = Z^,). It is then natural to study the operators acting between L(/1?)(G) spaces, where G has a compact invariant neighborhood. The theory of L(i>?)(G) is developed and boundedness and compactness conditions of a nonclassical type are obtained. It is demonstrated that the results extend easily to a somewhat broader class of integral operators. Several known results are strengthened or extended as incidental consequences of the investigation. Introduction. Convolution by/inLX(R) defines a bounded operator, Cy, on Lp(R) for 1 < p < oo; likewise, pointwise multiplication by A in LX(R) defines a bounded operator Ph on Lp(R). Excepting trivial cases, these operators are never compact. The composition PhCy of two such operators, which we term a productconvolution (PC) operator, is frequently compact. Asking exactly when this occurs motivates this paper. PC operators on Lp of a locally compact group arise in many areas of analysis. In [2] and [3] the compactness of certain PC operators was used to study induced representations of locally compact groups. PC operators (and their adjoint CP operators) arise naturally in many applied problems.Even with A in L^R) and/ in LX(R), we find that the "mixed-norm" spaces of [1] must be introduced to solve the compactness problem for PhC¡. These spaces also arise unavoidably when one attempts to weaken conditions on A and / and keep PhCj bounded. Various mixed-norm conditons on A and/guarantee boundedness of PhCf from Lp to Lq and, in fact, between mixed-norm spaces. Conversely, PhCj may be bounded from Lp to Lq (or from one mixed-norm space to another) with neither A nor/ in any Lr, but both A and/ must belong to mixed-norm spaces. Thus mixed-norm spaces are the natural setting for studying bounded PC operators. We treat a wide class of PC operators and, in all but one case, find necessary and sufficient conditions for compactness. The conditions involve only membership in a mixed-norm space or, in one instance, translational continuity in mixednorm spaces; they are usually easily checked for explicit examples. By applying various manipulations to PC operators, we develop easily applied necessary and sufficient conditions for compactness of a wider variety of integral operators which

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Weighted local Hardy spaces with variable exponents

Izuki

Nogayama

Noi

et al. 2023

Mathematische Nachrichten

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This paper defines local weighted Hardy spaces with variable exponents. Local Hardy spaces permit atomic decomposition, which is one of the main themes in this paper. A consequence is that the atomic decomposition is obtained for the functions in the Lebesgue spaces with exponentially decaying exponent. As applications, we obtain the boundedness of singular integral operators, the Littlewood–Paley characterization, and wavelet decomposition.

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Unions et intersections d'espaces $L^p$ invariantes par translation ou convolution

Cited by 39 publications

References 11 publications

A characterization of minimal homogeneous Banach spaces

A characterization of minimal homogeneous Banach spaces

Product-Convolution Operators and Mixed-Norm Spaces

Weighted local Hardy spaces with variable exponents

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