By Sagun Chanillo, Bruno Franchi, Guozhen Lu, Carlos Perez, Eric T. Sawyer
This choice of articles and surveys is dedicated to Harmonic research, comparable Partial Differential Equations and purposes and particularly to the fields of analysis to which Richard L. Wheeden made profound contributions. The papers care for Weighted Norm inequalities for classical operators like Singular integrals, fractional integrals and maximal services that come up in Harmonic research. different papers take care of purposes of Harmonic research to Degenerate Elliptic equations, variational difficulties, numerous complicated variables, capability concept, unfastened limitations and boundary habit of functions.
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Additional info for Harmonic Analysis, Partial Differential Equations and Applications: In Honor of Richard L. Wheeden
0 ; ı; M1 / if there exists a measure that is doubling on ı-balls (d is assumed to be symmetric there, however by similar technique, one could get the same conclusion with nonsymmetric quasimetric). 8] where it is assumed D which is doubling on all ı-balls and q p there. 11]. However, the quasimetric has been assumed to be symmetric there. (4) In quasimetric spaces, the constant in (11) and (12) is only known to be > 1 when balls are not known to be Boman domains. z; x/sp Ã1=p ; (16) we will then obtain fractional Sobolev inequalities.
00100 17. R. L. Torrea, Extension problem and Harnack’s inequality for some fractional operators. Commun. Partial Differ. Equ. R2 /. R2 /. x; t/ 2 R; here is the kinematic viscosity coefficient. Note we have divided the Navier–Stokes equation by the constant density of the fluid and thus in our equation is the dynamic viscosity coefficient divided by the density, assumed constant. e. derivative in the x variables). We also sometimes use the notation @x to denote a derivative in the x variables when we have no need to be specific which space variable we are differentiating in.
Is a vector field in R2 with vanishing divergence. The desired conclusion then follows from (2) and the two-dimensional result of Bourgain and Brezis  (see also ,  and ). Since the proof of the two-dimensional result of Bourgain and Brezis  is actually quite simple, we adapt it here in our particular setting, for the convenience of the reader. The main point here is that if ! v1 ; v2 / WD . R2 / the last inequality following from an application of Hardy’s inequality (alternatively, y 1 one can see that the last inequality holds, by writing jyj as r jyj , and by integrating by parts).