Adjoint Brascamp-Lieb inequalities | What’s new

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Adjoint Brascamp-Lieb inequalities | What’s new

keiseronlineuniversity.com

7 August 2023

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Jon Bennett and I’ve simply uploaded to the arXiv our paper “Adjoint Brascamp-Lieb inequalities“. On this paper, we observe that the household of multilinear inequalities referred to as the Brascamp-Lieb inequalities (or Holder-Brascamp-Lieb inequalities) admit an adjoint formulation, and discover the speculation of those adjoint inequalities and a few of their penalties.

To inspire issues allow us to evaluate the classical principle of adjoints for linear operators. If one has a bounded linear operator ${T: L^p(X) rightarrow L^q(Y)}$ for some measure areas ${X,Y}$ and exponents ${1 < p, q < infty}$ , then one can outline an adjoint linear operator ${T^*: L^{q'}(Y) rightarrow L^{p'}(X)}$ involving the twin exponents ${frac{1}{p}+frac{1}{p'} = frac{1}{q}+frac{1}{q'} = 1}$ , obeying (formally at the very least) the duality relation

$displaystyle langle Tf, g rangle = langle f, T^* g rangle (1)$

for appropriate take a look at capabilities ${f, g}$ on ${X, Y}$ respectively. Utilizing the twin characterization

$displaystyle |f|_{L^{p'}(X)} = sup_{g: |g|_{L^p(X)} leq 1} |langle f, g rangle|$

of ${L^{p'}(X)}$ (and equally for ${L^{q'}(Y)}$ ), one can present that ${T^*}$ has the identical operator norm as ${T}$ .

There’s a barely completely different strategy to proceed utilizing Hölder’s inequality. For sake of exposition allow us to make the simplifying assumption that ${T}$ (and therefore additionally ${T^*}$ ) maps non-negative capabilities to non-negative capabilities, and ignore problems with convergence or division by zero within the formal calculations under. Then for any affordable perform ${g}$ on ${Y}$ , we’ve

$displaystyle | T^* g |_{L^{p'}(X)}^{p'} = langle (T^* g)^{p'-1}, T^* g rangle = langle T (T^* g)^{p'-1}, g rangle$

$displaystyle leq |T|_{op} |(T^* g)^{p'-1}|_{L^p(X)} |g|_{L^{p'}(Y)}$

$displaystyle = |T|_{op} |T^* g |_{L^{p'}(X)}^{p'-1} |g|_{L^{p'}(Y)};$

by (1) and Hölder; dividing out by ${|T^* g |_{L^{p'}(X)}^{p'-1}}$ we get hold of ${|T^*|_{op} leq |T|_{op}}$ , and an analogous argument additionally recovers the reverse inequality.

The primary argument additionally extends to some extent to multilinear operators. As an illustration if one has a bounded bilinear operator ${B: L^p(X) times L^q(Y) rightarrow L^r(Z)}$ for ${1 < p,q,r < infty}$ then one can then outline adjoint bilinear operators ${B^{*1}: L^q(Y) times L^{r'}(Z) rightarrow L^{p'}(X)}$ and ${B^{*2}: L^p(X) times L^{r'}(Z) rightarrow L^{q'}(Y)}$ obeying the relations

$displaystyle langle B(f, g),h rangle = langle B^{*1}(g,h), f rangle = langle B^{*2}(f,h), g rangle$

and with precisely the identical operator norm as ${B}$ . Additionally it is doable, formally at the very least, to adapt the Hölder inequality argument to succeed in the identical conclusion.

On this paper we observe that the Hölder inequality argument might be modified within the case of Brascamp-Lieb inequalities to acquire a distinct kind of adjoint inequality. (Steady) Brascamp-Lieb inequalities take the shape

$displaystyle int_{{bf R}^d} prod_{i=1}^k f_i^{c_i} circ B_i leq mathrm{BL}(mathbf{B},mathbf{c}) (prod_{i=1}^k int_{{bf R}^{d_i}} f_i)^{c_i}$

for numerous exponents ${c_1,dots,c_k}$ and surjective linear maps ${B_i: {bf R}^d rightarrow {bf R}^{d_i}}$ , the place ${f_i: {bf R}^{d_i} rightarrow {bf R}}$ are arbitrary non-negative measurable capabilities and ${mathrm{BL}(mathbf{B},mathbf{c})}$ is the very best fixed for which this inequality holds for all such ${f_i}$ . [There is also another inequality involving variances with respect to log-concave distributions that is also due to Brascamp and Lieb, but it is not related to the inequalities discussed here.] Well-known examples of such inequalities embody Hölder’s inequality and the sharp Younger convolution inequality; one other is the Loomis-Whitney inequality, the primary non-trivial instance of which is

$displaystyle int_{{bf R}^3} f(y,z)^{1/2} g(x,z)^{1/2} h(x,y)^{1/2}$

$displaystyle leq (int_{{bf R}^2} f)^{1/2} (int_{{bf R}^2} g)^{1/2} (int_{{bf R}^2} h)^{1/2} (2)$

for all non-negative measurable ${f,g,h: {bf R}^2 rightarrow {bf R}}$ . There are additionally discrete analogues of those inequalities, through which the Euclidean areas ${{bf R}^d, {bf R}^{d_i}}$ are changed by discrete abelian teams, and the surjective linear maps ${B_i}$ are changed by discrete homomorphisms.

The operation ${f mapsto f circ B_i}$ of pulling again a perform on ${{bf R}^{d_i}}$ by a linear map ${B_i: {bf R}^d rightarrow {bf R}^{d_i}}$ to create a perform on ${{bf R}^d}$ has an adjoint pushforward map ${(B_i)_*}$ , which takes a perform on ${{bf R}^d}$ and principally integrates it on the fibers of ${B_i}$ to acquire a “marginal distribution” on ${{bf R}^{d_i}}$ (presumably multiplied by a normalizing determinant issue). The adjoint Brascamp-Lieb inequalities that we get hold of take the shape

$displaystyle |f|_{L^p({bf R}^d)} leq mathrm{ABL}( mathbf{B}, mathbf{c}, theta, p) prod_{i=1}^k |(B_i)_* f |_{L^{p_i}({bf R}^{d_i})}^{theta_i}$

for non-negative ${f: {bf R}^d rightarrow {bf R}}$ and numerous exponents ${p, p_i, theta_i}$ , the place ${mathrm{ABL}( mathbf{B}, mathbf{c}, theta, p)}$ is the optimum fixed for which the above inequality holds for all such ${f}$ ; informally, such inequalities management the ${L^p}$ norm of a non-negative perform by way of its marginals. It seems that each Brascamp-Lieb inequality generates a household of adjoint Brascamp-Lieb inequalities (with the exponent ${p}$ being much less than or equal to ${1}$ ). As an illustration, the adjoints of the Loomis-Whitney inequality (2) are the inequalities

$displaystyle | f |_{L^p({bf R}^3)} leq | (B_1)_* f |_{L^{p_1}({bf R}^2)}^{theta_1} | (B_2)_* f |_{L^{p_2}({bf R}^2)}^{theta_2} | (B_3)_* f |_{L^{p_3}({bf R}^2)}^{theta_3}$

for all non-negative measurable ${f: {bf R}^3 rightarrow {bf R}}$ , all ${theta_1, theta_2, theta_3>0}$ summing to ${1}$ , and all ${0 < p leq 1}$ , the place the ${p_i}$ exponents are outlined by the method

$displaystyle frac{1}{2} (1-frac{1}{p}) = theta_i (1-frac{1}{p_i})$

and the ${(B_i)_* f:{bf R}^2 rightarrow {bf R}}$ are the marginals of ${f}$ :

$displaystyle (B_1)_* f(y,z) := int_{bf R} f(x,y,z) dx$

$displaystyle (B_2)_* f(x,z) := int_{bf R} f(x,y,z) dy$

$displaystyle (B_3)_* f(x,y) := int_{bf R} f(x,y,z) dz.$

One can derive these adjoint Brascamp-Lieb inequalities from their ahead counterparts by a model of the Hölder inequality argument talked about beforehand, at the side of the statement that the pushforward maps ${(B_i)_*}$ are mass-preserving (i.e., they protect the ${L^1}$ norm on non-negative capabilities). Conversely, it seems that the adjoint Brascamp-Lieb inequalities are solely out there when the ahead Brascamp-Lieb inequalities are. Within the discrete case the ahead and adjoint Brascamp-Lieb constants are basically equivalent, however within the steady case they’ll (and sometimes do) differ by as much as a relentless. Moreover, whereas within the ahead case there’s a well-known theorem of Lieb that asserts that the Brascamp-Lieb constants might be computed by optimizing over gaussian inputs, the identical assertion is barely true as much as constants within the adjoint case, and actually typically the gaussians will fail to optimize the adjoint inequality. The state of affairs seems to be sophisticated; roughly talking, the adjoint inequalities solely use a portion of the vary of doable inputs of the ahead Brascamp-Lieb inequality, and this portion typically misses the gaussian inputs that will in any other case optimize the inequality.

Now we have situated a modest variety of functions of the adjoint Brascamp-Lieb inequality (however hope that there will likely be extra sooner or later):

We additionally document a variety of numerous of the adjoint Brascamp-Lieb inequalities, together with discrete variants, and a reverse inequality involving ${L^p}$ norms with ${p>1}$ somewhat than ${p<1}$ .

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