Intersections of Continuous, Lipschitz, Holder Class, and Smooth Functions

1995 ◽  
Vol 123 (4) ◽  
pp. 1157
Author(s):  
Jack B. Brown
Keyword(s):  
Smooth Functions ◽  
Hölder Class ◽  
Holder Class

MIXED FRACTIONAL DIFFERENTIATION OPERATORS IN HÖLDER SPACES DEFINED BY USUAL HÖLDER CONDITION

2019 ◽  
Author(s):  
T. Mamatov ◽  
R. Sabirova ◽  
D. Barakaev
Keyword(s):  
Fractional Derivative ◽  
Fractional Differentiation ◽  
Hölder Condition ◽  
Hölder Spaces ◽  
Main Interest ◽  
Hölder Class ◽  
Holder Class ◽  
Function Of Two Variables ◽  
Holder Spaces

We study mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The main interest being in the evaluation of the latter for the mixed fractional derivative in the cases Hölder class defined by usual Hölder condition

scholarly journals Weighted Morrey Spaces Related to Certain Nonnegative Potentials and Riesz Transforms

2019 ◽  
Vol 2019 ◽  
pp. 1-18
Author(s):  
Hua Wang
Keyword(s):  
Riesz Transform ◽  
Morrey Spaces ◽  
Riesz Transforms ◽  
Strong Type ◽  
Type Estimate ◽  
Reverse Hölder Class ◽  
Hölder Class ◽  
Weighted Morrey Spaces ◽  
Holder Class ◽  
Reverse Hölder

LetL=-Δ+Vbe a Schrödinger operator, whereΔis the Laplacian onRdand the nonnegative potentialVbelongs to the reverse Hölder classRHqforq≥d. The Riesz transform associated with the operatorL=-Δ+Vis denoted byR=∇(-Δ+V)-1/2and the dual Riesz transform is denoted byR⁎=(-Δ+V)-1/2∇. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder classRHqforq≥d. Then we will establish the mapping properties of the operatorRand its adjointR⁎on these new spaces. Furthermore, the weighted strong-type estimate and weighted endpoint estimate for the corresponding commutators[b,R]and[b,R⁎]are also obtained. The classes of weights, classes of symbol functions, and weighted Morrey spaces discussed in this paper are larger thanAp,BMO(Rd), andLp,κ(w)corresponding to the classical Riesz transforms (V≡0).

Oscillatory singular integral operators with Hölder class kernels on Hardy spaces

2019 ◽  
Vol 31 (2) ◽  
pp. 535-542
Author(s):  
Yibiao Pan
Keyword(s):  
Singular Integral ◽  
Hardy Spaces ◽  
Singular Integrals ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Oscillatory Singular Integral ◽  
Hölder Class ◽  
Holder Class ◽  
Oscillatory Singular Integrals

AbstractA sharp logarithmic bound is established for the {H^{1}}-norm of oscillatory singular integrals with quadratic phases and Hölder class kernels. Prior results had relied on a {C^{1}}-assumption on the kernel.

scholarly journals Parabolic Fractional Maximal Operator in Modified Parabolic Morrey Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Vagif S. Guliyev ◽  
Kamala R. Rahimova
Keyword(s):  
Maximal Operator ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Reverse Hölder Class ◽  
Fractional Maximal Operator ◽  
Hölder Class ◽  
Holder Class ◽  
Homogeneous Dimension ◽  
Reverse Hölder ◽  
Limiting Case

We prove that the parabolic fractional maximal operatorMαP,0≤α<γ, is bounded from the modified parabolic Morrey spaceM̃1,λ,P(ℝn)to the weak modified parabolic Morrey spaceWM̃q,λ,P(ℝn)if and only ifα/γ≤1-1/q≤α/(γ-λ)and fromM̃p,λ,P(ℝn)toM̃q,λ,P(ℝn)if and only ifα/γ≤1/p-1/q≤α/(γ-λ). Hereγ=trPis the homogeneous dimension onℝn. In the limiting case(γ-λ)/α≤p≤γ/αwe prove that the operatorMαPis bounded fromM̃p,λ,P(ℝn)toL∞(ℝn). As an application, we prove the boundedness ofMαPfrom the parabolic Besov-modified Morrey spacesBM̃pθ,λs(ℝn)toBM̃qθ,λs(ℝn). As other applications, we establish the boundedness of some Schrödinger-ype operators on modified parabolic Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.

Holomorphic Frames for Weakly Converging Holomorphic Vector Bundles

2006 ◽  
Vol 49 (1) ◽  
pp. 36-40 ◽  
Author(s):  
Georgios D. Daskalopoulos ◽  
Richard A. Wentworth
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Vector Bundles ◽  
Convergent Sequence ◽  
Complex Vector Bundle ◽  
Complex Vector ◽  
Hölder Class ◽  
Holomorphic Vector Bundles ◽  
Holder Class

AbstractUsing a modification of Webster's proof of the Newlander–Nirenberg theorem, it is shown that, for a weakly convergent sequence of integrable unitary connections on a complex vector bundle over a complex manifold, there is a subsequence of local holomorphic frames that converges strongly in an appropriate Holder class.

scholarly journals The Boundedness of Marcinkiewicz Integrals Associated with Schrödinger Operator on Morrey Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Dongxiang Chen ◽  
Fangting Jin
Keyword(s):  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Morrey Spaces ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Marcinkiewicz Integrals ◽  
Reverse Hölder Class ◽  
Hölder Class ◽  
Holder Class ◽  
Reverse Hölder

LetL=-Δ+Vbe a Schrödinger operator, whereVbelongs to some reverse Hölder class. The authors establish the boundedness of Marcinkiewicz integrals associated with Schrödinger operators and their commutators on Morrey spaces.

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