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Adjoint Brascamp-Lieb inequalities | What’s new

Adjoint Brascamp-Lieb inequalities | What’s new

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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 idea of those adjoint inequalities and a few of their penalties.

To inspire issues allow us to evaluation the classical concept 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 no less than) the duality relation

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

for appropriate take a look at features {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 approach 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 features to non-negative features, and ignore problems with convergence or division by zero within the formal calculations beneath. Then for any cheap 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^q(Y)}

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

by (1) and Hölder; dividing out by {|T^* g |_{L^{p'}(X)}^{p'-1}} we acquire {|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 example 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}. It is usually potential, formally no less than, to adapt the Hölder inequality argument to succeed in the identical conclusion.

On this paper we observe that the Hölder inequality argument could be modified within the case of Brascamp-Lieb inequalities to acquire a special 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 features 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 embrace 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, by 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}} (probably multiplied by a normalizing determinant issue). The adjoint Brascamp-Lieb inequalities that we acquire 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 when it comes to 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 example, 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 components

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, along side the commentary that the pushforward maps {(B_i)_*} are mass-preserving (i.e., they protect the {L^1} norm on non-negative features). Conversely, it seems that the adjoint Brascamp-Lieb inequalities are solely accessible 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 will (and sometimes do) differ by as much as a continuing. Moreover, whereas within the ahead case there’s a well-known theorem of Lieb that asserts that the Brascamp-Lieb constants could be computed by optimizing over gaussian inputs, the identical assertion is simply true as much as constants within the adjoint case, and actually usually the gaussians will fail to optimize the adjoint inequality. The state of affairs seems to be difficult; roughly talking, the adjoint inequalities solely use a portion of the vary of potential inputs of the ahead Brascamp-Lieb inequality, and this portion usually misses the gaussian inputs that might in any other case optimize the inequality.

We’ve got positioned a modest variety of functions of the adjoint Brascamp-Lieb inequality (however hope that there might be extra sooner or later):

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

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