The Structure of Functions

Abstract

Many of the original research and survey monographs in pure and applied mathematics published by Birkhäuser in recent decades have been groundbreaking and have come to be regarded as foundational to the subject. Through the MBC Series, a select number of these modern classics, entirely uncorrected, are being re-released in paperback (and as eBooks) to ensure that these treasures remain accessible to new generations of students, scholars, and researchers.

Quasilinear and Hessian equations of Lane–Emden type

“…By the refined localization principle on the smooth domain Ω for the function space W p, q q−p+1 (see, e.g., [Tri,Th. 5.14]) we have…”

Nonlinear approximation with general wave packets

“…Let us briefly recall the definition of the homogeneous TriebelLizorkin and Besov spaces (see also, e.g. [18]). …”

Local growth envelopes of spaces of generalized smoothness: the subcriticalcase

“…In the case σ p < s < n/p, which we want to study here (the so-called sub-critical case), we have just pointed out that we always have that A . This motivates the definition below, which tries to make more precise the corresponding idea of Haroske [8] and Triebel [25]. if u is the minimum (assuming that it exists) of all v > 0 such that …”

“…Recently, Haroske [8] and Triebel [25] studied the type of essential unboundedness of functions in such spaces which are not embedded in L ∞ , and with the further restriction s > n(1/p − 1) + for the parameters (basically this guarantees one is dealing with regular distributions). The unboundedness was measured by the determination of the so-called local growth envelope function E LG F s pq or E LG B s pq which we shall write as E LG A s pq , for short , which is a (preferably) continuous representative in the equivalence class of all positive decreasing functions which are equivalent to E LG |A s pq (t) := sup f * (t) : f A s pq ≤ 1 in some interval (0, ε] (for some ε ∈ (0, 1]).…”