Riesz transforms on variable Lebesgue spaces with Gaussian measure

Abstract

We give sufficient conditions on variable exponent functions p : R n → [1, ∞) for which the higher-order Riesz transforms, associated with the Ornstein-Uhlenbeck semigroup, are bounded on L p(•) (R n , dγ ), where γ denotes the Gaussian measure.

Harmonic analysis operators associated with Laguerre polynomial expansions on variable Lebesgue spaces

“…In [15], Dalmasso and Scotto studied Riesz transforms in the Gaussian setting on variable Lebesgue spaces. In order to do this, they introduced a new class of exponents which is contained in LH…”

“…). Main properties of the functions in P ∞ e (R n ) were established in [15]. Maximal operators defined by the heat semigroup ( [28]) and Riesz type singular integrals ( [16] and [31]) associated with the Ornstein-Uhlenbeck differential operator were studied on L p(•) (R n , γ n ) with p ∈ LH 0 (R n ) ∩ P ∞ e (R n ), where dγ n denotes the Gaussian measure.…”

The Boundedness of the Ornstein-Uhlenbeck semigroup on variable Lebesgue spaces with respect to the Gaussian measure

“…In this section we are going to consider Lebesgue variable spaces with respect to the Gaussian measure γ d , L p(•) (R d , γ d ). The next condition was introduced by E. Dalmasso and R. Scotto in [6].…”

The Boundedness of General Alternative Singular Integrals with Respect to the Gaussian Measure

“…In an forthcoming paper [6], following [2], we prove that the general alternative Gaussian singular integrals T F,m are also continuous on Gaussian variable Lebesgue spaces under a condition of regularity on p(•).…”