Abstract
The convolution of two weighted balls of measures is proved to be contained in a third weighted ball if and only if the supremal convolution of the corresponding two weights is less than or equal to the third weight. Here supremal convolution is introduced as a type of convolution in which integration is replaced with supremum formation. Invoking duality the equivalence implies a characterization of equicontinuity of weight-bounded sets of convolution operators having weighted spaces of continuous functions as domain and range. The overall result is a constructive method to define weighted spaces on which a given set of convolution operators acts as an equicontinuous family of endomorphisms. The result is applied to linear combinations of fractional Weyl integrals and derivatives with orders and coefficients from a given bounded set.
Highlights
Necessary and sufficient conditions are known, that make fractional integration a bounded operator between weighted (Lebesgue) spaces, applications, extensions and generalizations of weighted convolution inequalities continue to attract widespread interest in potential analysis and its applications to partial differential equations [3,6,13,14,15,16,17,21,27,35,36]
Deviating from the traditional focus on Banach spaces and fractional Riesz integrals, the present article studies weighted convolution algebras of Radon measures and fractional Weyl integrals operating as equicontinuous families of linear endomorphisms on weighted locally convex spaces of continuous functions
The main objective of this article is to introduce and prove certain weighted norm inequalities for convolution operators given in our Theorems 5 and 6
Summary
Necessary and sufficient conditions are known, that make fractional integration a bounded operator between weighted (Lebesgue) spaces, (see [25,28,31,32,33] and references therein), applications, extensions and generalizations of weighted convolution inequalities continue to attract widespread interest in potential analysis and its applications to partial differential equations [3,6,13,14,15,16,17,21,27,35,36]. Cv(W ) denotes the set of continuous functions on G with w| f | vanishing at infinity for all w ∈ W and TW is the locally convex topology generated by the weighted supremum norms f → f w := sup{w(x)| f (x)| : x ∈ G}, w ∈ W. Requires to drop the positivity assumption on the weights in (1.6) This forces us to deal with convolutes (μf )(x) that can diverge for some or even all x ∈ G, and in turn, to deal with integrals of C∞-valued measurable functions that are allowed to diverge. Weyl integrals and derivatives as linear endomorphisms on weighted function spaces
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