Sup-norm-closable bilinear forms and Lagrangians

Abstract

Abstract. We consider symmetric non-negative definite bilinear forms on algebras of bounded real valued functions and investigate closability with respect to the supremum norm. In particular, any Dirichlet form gives rise to a sup-norm closable bilinear form. Under mild conditions a sup-norm closable bilinear form admits finitely additive energy measures. If, in addition, there exists a (countably additive) energy dominant measure, then a sup-norm closable bilinear form can be turned into a Dirichlet form admitting a carré du champ. Moreover, we can always transfer the bilinear form to an isometrically isomorphic algebra of bounded functions on the Gelfand spectrum, where these measures exist. Our results complement a former closability study of Mokobodzki for the locally compact and separable case.

Densely defined non‐closable curl on carpet‐like metric measure spaces

“…By the arguments of [, Lemma 1.4.2 and Problem 1.4.1] the algebra provides a special standard core for . Combining [, Corollary 7.1.2] and [, Corollary 8.3] we obtain .…”

“…Our paper is a part of a broader program that aims to connect research on derivatives on fractals ( [8,9,[13][14][15][25][26][27][28]31,33,35,[37][38][39][40][41]46,51,54] and references therein) and on more general regular Dirichlet spaces [29,30,34] with classical and geometric analysis on metric measure spaces ( [6,10,12,[21][22][23][24][42][43][44][45]50] and references therein). In our previous article [36] we showed that on certain topologically one-dimensional spaces with a strongly local regular Dirichlet form one can prove a natural version of the Hodge theorem for 1-forms defined in 2 -sense: the set of harmonic 1-forms is dense in the orthogonal complement of the exact 1-forms.…”

“…In particular the reader can consult the papers [7,10,[42][43][44][45] and references therein. Our current paper is a step in a long-term program (see [29][30][31][32][33][34][35][36][37][38]) to develop parts of differential geometry and their applications to mathematical physics (see [3][4][5]) for spaces that carry diffusion processes but no other smooth structure. Our approach is somewhat complementary to the celebrated works [12,21,22] because, although our spaces are metrizable, we do not use any particular metric in an essential way, and we do not use functional inequalities.…”

Fractal snowflake domain diffusion with boundary and interior drifts

“…Figure 6 displays this most extreme case. Using (24) and the triangle inequality we can then see that the maximal difference of the values on f on these points is bounded by 4 −n 7 L E ∂Ω (f ). By averaging then also the difference of g n (x) and g n (x ) is bounded by that number.…”

“…Remark 6.2. In some sense Corollary 6.1 and formula (24) imply not only that g is Euclidean-Lipschitz, but also that g has "zero (Euclidean) tangential derivative" along ∂Ω, because an E ∂Ω -intrinsic Lipschitz function has to grow "infinitely slow" in the "tangential direction" to the Koch curve.…”