Sharp local uncertainty inequalities

Weighted Norm Inequalities for the Local Sharp Maximal Function

Weighted Rearrangement Inequalities for Local Sharp Maximal Functions

Several weighted rearrangement inequalities for uncentered and centered local sharp functions are proved. These results are applied to obtain new weighted weak-type and strong-type estimates for singular integrals. A self-improving property of sharp function inequalities is established.

Sharp Boundary Trace Inequalities

This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region Ω ⊂ R . The inequalities bound (semi-)norms of the boundary trace by certain norms of the function and its gradient on the region and two specific constants kρ and kΩ associated with the domain and a weight function. These inequalities are sharp in that there are functions for which eq...

Sharp Jackson inequalities

For trigonometric polynomials on [− , ] ≡ T , the classical Jackson inequalityEn(f )p C r (f, 1/n)p was sharpened by M. Timan for 1<p<∞ to yield n−r { n ∑ k=1 ksr−1Ek(f )p }1/s C r (f, n−1)p where s =max(p, 2). In this paper a general result on the relations between systems or sequences of best approximation and appropriate measures of smoothness is given. Approximation by algebraic polynomials...

SHARP AFFINE Lp SOBOLEV INEQUALITIES

In this paper we prove a sharp affine Lp Sobolev inequality for functions on R. The new inequality is significantly stronger than (and directly implies) the classical sharp Lp Sobolev inequality of Aubin [A2] and Talenti [T], even though it uses only the vector space structure and standard Lebesgue measure on R. For the new inequality, no inner product, norm, or conformal structure is needed at...