On the Differentiability of Lipschitz-Besov Functions (original) (raw)

On Some Properties of Functions from a Besov-Morrey Type Spaces

2018

In this paper it is constructed a new Besov-Morrey type spaces. Utilizing integral representation of generalized derivatives of functions defined on n-dimensional domains satisfying flexible φ-horn condition an embedding theorem is proved. Also, it is proved that the generalized derivatives of functions from this spaces satisfies the generalized Hölder condition. In this paper we introduce a Besov-Morrey type spaces B p,θ,φ,β(Gφ) and studied differential and differential-difference properties of functions from this spaces. Note that the spaces with parameters based on isotropic Sobolev space, while some I.Ross private values of indexes for the first time studied in the works of Ch.Morrey [8]. These results further developed and compiled in the works of V.P. Il’yin [6], Yu.V. Netrusov [13], V.S. Guliyev [3], Y.Sawano[15], V.Kokilashvili, A. Meskhi and H. Rafeiro [7], I. Ross[14] and [1, 10, 11, 12] and others. Let G be a domain in R and let φ(t) = (φ1(t), . . . , φn(t)), φj(t) > 0...

Continuity characterisations of differentiability of locally Lipschitz functions

Bulletin of the Australian Mathematical Society, 1990

Recently David Preiss contributed a remarkable theorem about the differentiability of locally Lipschitz functions on Banach spaces which have an equivalent norm differentiable away from the origin. Using his result in conjunction with Frank Clarke's non-smooth analysis for locally Lipschitz functions, continuity characterisations of differentiability can be obtained which generalise those for convex functions on Banach spaces. This result gives added information about differentiability properties of distance functions.

A remark on differentiable functions with partial derivatives in L p

Journal of Mathematical Analysis and Applications, 2004

We consider a definition of p,δp,δ-variation for real functions of several variables which gives information on the differentiability almost everywhere and the absolute integrability of its partial derivatives on a measurable set. This definition of p,δp,δ-variation extends the definition of n-variation of Malý and the definition of p-variation of Bongiorno. We conclude with a result of change of variables based on coarea formula.

Besov-Lipschitz and mean Besov-Lipschitz spaces of holomorphic functions on the unit ball

2011

We give several characterizations of holomorphic mean Besov-Lipschitz space on the unit ball in C N and appropriate Besov-Lipschitz space and prove the equivalences between them. Equivalent norms on the mean Besov-Lipschitz space involve different types of L p -moduli of continuity, while in characterizations of Besov-Lipschitz space we use not only the radial derivative but also the gradient and the tangential derivatives. The characterization in terms of the best approximation by polynomials is also given.

Rectangular differentiation of integrals of Besov functions

Mathematical Research Letters, 2002

We study the differentiation of integrals of functions in the Besov spaces B α,1 p (R n ), α > 0, 1 ≤ p < ∞, with respect to the basis of arbitrarily oriented rectangular parallelepipeds in R n . We show that positive results hold if α ≥ n−1 p and we give counterexamples for the case 0 < α < n p − 1. Similar results hold for B α,q p (R n ), q > 1. For more general bases we can also prove negative results for n p − 1 ≤ α < n−1 p .

On fine differentiability properties of Sobolev functions

Cornell University - arXiv, 2022

We study fine differentiability properties of functions in Sobolev spaces. We prove that the difference quotient of f ∈ W 1 p (R n) converges to the formal differential of this function in the W 1 p,loc-topology cap p-a. e. under an additional assumption of existence of a refined weak gradient. This result is extended to convergence of remainders in the corresponding Taylor formula for functions in W k p spaces. In addition, we prove Lp−approximately differentiability cap p −a.e for functions f ∈ W 1 p (R n) with a refined weak gradient.