Inverting thep-harmonic operator

Strong $A_{\infty}$-weights and scaling invariant Besov capacities

“…As a consequence of our result, (see [18], [11]), we can obtain strong A ∞ -weights of the form w = e nu , where u is a distributional solution of −div(|∇u| n−2 ∇u) = µ whenever µ is a signed Radon measure with small total variation. Indeed, it follows from the results of [18] and [11] that every distributional solution u of the previous equation has the property that the weak L n -norm of the distributional gradient of u is controlled by (|µ|(R n )) 1 n−1 .…”

On the Product of Functions in BMO and H^\text{1}

“…Recently a viable theory of existence and improved regularity for solutions of PDEs where the uniform ellipticity is lost, has been built out of the distributional div-curl products and null Lagrangians [25,31] . Second, these investigations bring us to new classes of functions, distributions and measures [32] , just to mention the grand L p -spaces [29,24,53].…”

“…Recent investigations of so-called very weak solutions of nonlinear PDEs [30,24] rely on these new classes of functions. Third, it seems likely that these methods will shed new light on harmonic analysis with more practical applications.…”

An essay on the Interpolation Theorem of Józef Marcinkiewicz – Polish patriot

“…The interested reader may wish to take a note of the interpolation lemmas in [1]. In the present paper I will try to elucidate some new advances of Marcinkiewicz interpolation theorem which arise from a study of the nonlinear p-harmonic type PDEs, [12,13,14,16,18,19,20]. The principal result in this paper can be described as follows:…”