A generalized Itô formula for time dependent functions of two-dimensional continuous semi-martingales is proved. The formula uses the local time of each coordinate process of the semi-martingale, left space and time first derivatives and second derivative ∇1 ∇ − 2 f only which are assumed to be of locally bounded variation in certain variables, and stochastic Lebesgue-Stieltjes integrals of two...

Let X be a space of homogeneous type in the sense of Coifman and Weiss and D a collection of balls in X . The authors introduce the localized atomic Hardy space H q D (X ) with p ∈ (0, 1] and q ∈ [1,∞] ∩ (p,∞], the localized Morrey-Campanato space E p D (X ) and the localized Morrey-Campanato-BLO space Ẽ p D (X ) with α ∈ R and p ∈ (0,∞) and establish their basic properties including H q D (X )...

We obtain global bounds in Lorentz-Morrey spaces for gradients of solutions to a class of quasilinear elliptic equations with low integrability data. The results are then applied to obtain sharp existence results in the framework of Morrey spaces for Riccati type equations with a gradient source term having growths below the natural exponent of the operator involved. A special feature of our re...

The dual purpose of this article is to establish bilinear Poincaré-type estimates associated to an approximation of the identity and to explore the connections between bilinear pseudo-differential operators and bilinear potential-type operators. The common underlying theme in both topics is their applications to Leibniz-type rules in Sobolev and Campanato-Morrey spaces under Sobolev scaling.

Currently published HOL formalizations of measure theory concentrate on the Lebesgue integral and they are restricted to realvalued measures. We lift this restriction by introducing the extended real numbers. We define the Borel σ-algebra for an arbitrary type forming a topological space. Then, we introduce measure spaces with extended real numbers as measure values. After defining the Lebesgue...

‎by considering a degenerate $p(x)-$laplacian equation‎, ‎a generalized compact embedding in weighted variable‎ ‎exponent sobolev space is presented‎. ‎multiplicity of positive solutions are discussed by applying fibering map approach for the‎ ‎corresponding nehari manifold‎.

We prove that the weak Morrey space WM is contained in $$M_{{q_1}}^p$$ for 1 ≤ q1 < q p ∞. As applications, we show if commutator [b, T] bounded from Lp to Lp,∞ some ∈ (1, ∞), then b BMO, where T a Calderón-Zygmund operator. Also, ∞, BMO and only [6, M . For belonging Lipschitz class, obtain similar results.

A singular operator with Cauchy kernel on the subspaces of weight Lebesgue space is considered. A sufficient condition for a bounded action of this operator from a subspace to another subspace of weight Lebesgue space of functions is found. These conditions are not identical withMuckenhoupt conditions. Moreover, the completeness, minimality, and basicity of sines and cosines systems are conside...