Variation of Calderón–Zygmund operators with matrix weight

Let [Formula: see text], [Formula: see text] and [Formula: see text] be a matrix [Formula: see text] weight. In this paper, we introduce a version of variation [Formula: see text] for matrix Calderón–Zygmund operators with modulus of continuity satisfying the Dini condition. We then obtain the [Formula: see text]-boundedness of [Formula: see text] with norm [Formula: see text] by first proving a sparse domination of the variation of the scalar Calderón–Zygmund operator, and then providing a convex body sparse domination of the variation of the matrix Calderón–Zygmund operator. The key step here is a weak type estimate of a local grand maximal truncated operator with respect to the scalar Calderón–Zygmund operator.

Compensation functions for factors of shifts of finite type

Let ${\it\pi}:X\rightarrow Y$ be an infinite-to-one factor map, where $X$ is a shift of finite type. A compensation function relates equilibrium states on $X$ to equilibrium states on $Y$. The $p$-Dini condition is given as a way of measuring the smoothness of a continuous function, with $1$-Dini corresponding to functions with summable variation. Two types of compensation functions are defined in terms of this condition. Given a fully supported invariant measure ${\it\nu}$ on $Y$, we show that the relative equilibrium states of a $1$-Dini function $f$ over ${\it\nu}$ are themselves fully supported, and have positive relative entropy. We then show that there exists a compensation function which is $p$-Dini for all $p>1$ which has relative equilibrium states supported by a subshift on which ${\it\pi}$ is a finite-to-one map onto $Y$.

Weighted inequalities for some integral operators with rough kernels

In this paper we study integral operators with kernels $$K(x,y) = k_1 (x - A_1 y) \cdots k_m \left( {x - A_m y} \right),$$ $$k_i \left( x \right) = {{\Omega _i \left( x \right)} \mathord{\left/ {\vphantom {{\Omega _i \left( x \right)} {\left| x \right|}}} \right. \kern-\nulldelimiterspace} {\left| x \right|}}^{{n \mathord{\left/ {\vphantom {n {q_i }}} \right. \kern-\nulldelimiterspace} {q_i }}}$$ where Ωi: ℝn → ℝ are homogeneous functions of degree zero, satisfying a size and a Dini condition, A i are certain invertible matrices, and n/q 1 +…+n/q m = n−α, 0 ≤ α < n. We obtain the appropriate weighted L p-L q estimate, the weighted BMO and weak type estimates for certain weights in A(p, q). We also give a Coifman type estimate for these operators.

Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spaces

In this article, we consider the Marcinkiewicz integrals with variable kernels defined byμΩ(f)(x)=(∫0∞|∫|x−y|≤tΩ(x,x−y)|x−y|n−1f(y)dy|2dtt3)1/2, whereΩ(x,z)∈L∞(ℝn)×Lq(Sn−1)forq> 1. We prove that the operatorμΩis bounded from Hardy space,Hp(ℝn), toLp(ℝn)space; and is bounded from weak Hardy space,Hp,∞(ℝn), to weakLp(ℝn)space formax{2n2n+1,nn+α}<p<1, ifΩsatisfies theL1,α-Dini condition with any0<α≤1.

On the size of the non-coincidence set of parabolic obstacle problems with applications to American option pricing

The following paper is devoted to the study of the positivity set $U=\{\mathcal{L}\phi>0\}$ arising in parabolic obstacle problems. It is shown that $U$ is contained in the non-coincidence set with a positive distance between the boundaries uniformly in the spatial variable if the boundary of $U$ satisfies an interior $C^1$-Dini condition in the space variable and a Lipschitz condition in the time variable. We apply our results to American option pricing and we thus show that the positivity set is strictly contained in the continuation region, which means that the option should not be exercised in $U$ or on the boundary of $U$.

Boundedness of higher-order Marcinkiewicz-Type integrals

LetAbe a function with derivatives of ordermandDγA∈Λ˙β(0<β<1,|γ|=m). The authors in the paper proved that ifΩ∈Ls(Sn−1) (s≥n/(n−β))is homogeneous of degree zero and satisfies a vanishing condition, then both the higher-order Marcinkiewicz-type integralμΩAand its variationμ˜ΩAare bounded fromLp(ℝn)toLq(ℝn)and fromL1(ℝn)toLn/(n−β),∞(ℝn), where1<p<n/βand1/q=1/p−β/n. Furthermore, ifΩsatisfies some kind ofLs-Dini condition, then bothμΩAandμ˜ΩAare bounded on Hardy spaces, andμΩAis also bounded fromLp(ℝn)to certain Triebel-Lizorkin space.

Singular Integrals in Weighted Lebesgue Spaces with Variable Exponent

In the weighted Lebesgue space with variable exponent the boundedness of the Calderón–Zygmund operator is established. The variable exponent 𝑝(𝑥) is assumed to satisfy the logarithmic Dini condition and the exponent β of the power weight ρ(𝑥) = |𝑥 – 𝑥0| β is related only to the value 𝑝(𝑥0). The mapping properties of Cauchy singular integrals defined on the Lyapunov curve and on curves of bounded rotation are also investigated within the framework of the above-mentioned weighted space.

Singular integrals and potentials in some Banach function spaces with variable exponent

We introduce a new Banach function space - a Lorentz type space with variable exponent. In this space the boundedness of singular integral and potential type operators is established, including the weighted case. The variable exponentp(t)is assumed to satisfy the logarithmic Dini condition and the exponentβof the power weightω(t)=|t|βis related only to the valuep(0). The mapping properties of Cauchy singular integrals defined on Lyapunov curves and on curves of bounded rotation are also investigated within the framework of the introduced spaces.