Convolution inequalities for Besov and Triebel--Lizorkin spaces, and applications to convolution semigroups

Abstract

We establish convolution inequalities for Besov spaces B s p,q and Triebel-Lizorkin spaces F s p,q . As an application, we study the mapping properties of convolution semigroups, considered as operators on the function spaces A s p,q , A ∈ {B, F }. Our results apply to a wide class of convolution semigroups including the Gauß-Weierstraß semigroup, stable semigroups and heat kernels for higher-order powers of the Laplacian (−∆) m , and we can derive various caloric smoothing estimates.The convolution f * g of two functions f, g ∶ R n → R is defined byprovided that the integral exists. It is well known that the convolution f * g inherits the regularity of both f and g, in the sense that if, say,In this article, we study the regularizing properties of the convolution in the scales of Besov spaces B s p,q and Triebel-Lizorkin spaces F s p,q . Having in mind that the parameters s and p describe the regularity, resp., integrability properties of f ∈ A s p,q , A ∈ {B, F }, it is natural to expect that the implication f ∈ A s p1,q1 , g ∈ A u

Yet another notion of irregularity through small ball estimates

“…We also recall a sewing lemma for finite variation paths (see for instance We shall use the above lemma to define the integral T 0 f(s, θ s − ω s ) ds for suitable paths (θ s ) s∈[0,T ] , (ω s ) s∈[0,T ] on R d , and non-smooth f : R d → R d . To this end we recall a standard result for convolutions (see for instance [KL21]). Theorem 6.2.…”

Regularization by random translation of potentials for the continuous PAM and related models in arbitrary dimension

“…We denote by C n the space of n-times continuously differentiable functions such that (1.11) is finite. Let us remark that the first two spaces above are related to more general Besov spaces in the sense that H α = B α 2,2 for any α ∈ R and C α = B α ∞,∞ for α > 0 and α ∈ N. In particular, note that by Young's inequality in Besov spaces [27], we have…”

Weak solutions for singular multiplicative SDEs via regularization by noise

“…By Young's inequality in Besov spaces [KS21], the spatial regularity of the averaging operator is therefore the sum of the regularity of b and L, yielding also a quantifiable regularization. Let us remark at this point that while the results in [GG21] are sharper, allowing even to consider non-autonomous equations, the approach in [HP21] has the merit of being pathwise stable in the sense Theorem 2.1 (see also Remark 2.4).…”